may overlap a bit with previous lecture(s)…

Mathematics demystified

January 9, 2019 – 4:24 am

may overlap a bit with previous lecture(s)…

November 13, 2018 – 3:16 pm

**Question I:**

Show that the equation of the tangent to the curve and can be represented in the form:

**Question 2:**

Show that the derivative of the function , when and , when vanishes on an infinite set of points of the interval $latex (0,1), Hint: Use Rolle’s theorem.

**Question 3:**

Prove that for . Use Lagrange’s theorem.

**Question 4:**

Find the largest term in the sequence . Hint: Consider the function in the interval .

**Question 5:**

A point P is given on the circumference of a circle with radius r. Chords QR are drawn parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

**Question 6:**

Find the polynomial of degree 6, which satisfies and has a local maximum at and local minimum at and 2.

**Question 7:**

For the circle , find the value of r for which the area enclosed by the tangents drawn from the point to the circle and the chord of contact is maximum.

**Question 8:**

Suppose that f has a continuous derivative for all values of x and , with for all x. Prove that .

**Question 9:**

Show that , if .

**Question 10:**

Let . Show that the equations has a unique root in the interval and identify it.

November 13, 2018 – 6:26 am

**Question 1.**

If the point on , where , where the tangent is parallel to has an ordinate , then what is the value of ?

**Question 2:**

Prove that the segment of the tangent to the curve , which is contained between the coordinate axes is bisected at the point of tangency.

**Question 3:**

Find all the tangents to the curve for that are parallel to the line .

**Question 4:**

Prove that the curves , where , and , where is a differentiable function have common tangents at common points.

**Question 5:**

Find the condition that the lines may touch the curve .

**Question 6:**

Find the equation of a straight line which is tangent to one point and normal to the point on the curve , and .

**Question 7:**

Three normals are drawn from the point to the curve . Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

**Question 8:**

If and are lengths of the perpendiculars from origin on the tangent and normal to the curve respectively, prove that .

**Question 9:**

Show that the curve , and is symmetrical about x-axis and has no real points for . If the tangent at the point t is inclined at an angle to OX, prove that . If the tangent at meets the curve again at Q, prove that the tangents at P and Q are at right angles.

**Question 10:
**

Find the condition that the curves and intersect orthogonality and hence show that the curves and also intersect orthogonally.

More later,

Nalin Pithwa.

October 19, 2018 – 6:33 pm

**“Easy” questions:**

Question 1:

Find the slope of the tangent to the curve represented by the curve and at the point .

Question 2:

Find the co-ordinates of the point P on the curve , the tangent at which is perpendicular to the line .

Question 3:

Find the co-ordinates of the point lying in the first quadrant on the ellipse so that the area of the triangle formed by the tangent at P and the co-ordinate axes is the smallest.

Question 4:

The function , where is

(a) increasing on

(b) decreasing on

(c) increasing on and decreasing on

(d) decreasing on and increasing on .

Fill in the correct multiple choice. Only one of the choices is correct.

Question 5:

Find the length of a longest interval in which the function is increasing.

Question 6:

Let , then is

(a) increasing on

(b) decreasing on

(c) increasing on

(d) decreasing on .

Fill in the correct choice above. Only one choice holds true.

Question 7:

Consider the following statements S and R:

S: Both and are decreasing functions in the interval .

R: If a differentiable function decreases in the interval , then its derivative also decreases in .

Which of the following is true?

(i) Both S and R are wrong.

(ii) Both S and R are correct, but R is not the correct explanation for S.

(iii) S is correct and R is the correct explanation for S.

(iv) S is correct and R is wrong.

Indicate the correct choice. Only one choice is correct.

Question 8:

For which of the following functions on , the Lagrange’s Mean Value theorem is not applicable:

(i) , when ; and , when .

(ii) , when ; and , when .

(iii)

(iv) .

Only one choice is correct. Which one?

Question 9:

How many real roots does the equation have?

Question 10:

What is the difference between the greatest and least values of the function ?

More later,

Nalin Pithwa.

December 18, 2017 – 8:43 pm

**Part I: Multiple Choice Questions:**

*Example 1:*

Locus of the mid-points of the chords of the circle which subtend a right angle at the centre is (a) (b) (c) (d)

*Answer 1: C.*

Solution 1:

Let O be the centre of the circle , and let AB be any chord of this circle, so that . Let be the mid-point of AB. Then, OM is perpendicular to AB. Hence, . Therefore, the locus of is .

*Example 2:*

If the equation of one tangent to the circle with centre at from the origin is , then the equation of the other tangent through the origin is (a) (b) (c) (d) .

*Answer 2: C.*

Solution 2:

Since touches the given circle, its radius equals the length of the perpendicular from the centre to the line . That is,

.

Let be the equation of the other tangent to the circle from the origin. Then,

, which gives two values of m and hence, the slopes of two tangents from the origin, with the product of the slopes being -1. Since the slope of the given tangent is -3, that of the required tangent is 1/3, and hence, its equation is .

*Example 3.*

A variable chord is drawn through the origin to the circle . The locus of the centre of the circle drawn on this chord as diameter is (a) (b) (c) (d) .

*Answer c.*

Solution 3:

Let be the centre of the required circle. Then, being the mid-point of the chord of the given circle, its equation is .

Since it passes through the origin, we have .

Hence, locus of is .

**Quiz problem:**

A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is (a) (b) (c) (d) .

*To be continued,*

*Nalin Pithwa.*

September 27, 2017 – 1:52 am

**Section I:**

**Definition of a Circle:**

A circle is the locus of a point which moves in a plane so that it’s distance from a *fixed point* in the plane is always *constant*.The fixed point is called the *centre* of the circle and the constant distance is called its *radius*.

**Section II:**

**Equations of a circle:**

- An equation of a circle with centre and radius r is .
- An equation of a circle with centre and radius r is .
- An equation of a circle on the line segment joining and as diameter is .
- General equation of a circle is : where g, f, and c are constants
- centre of this circle is
- Its radius is ,
- Length of the intercept made by this circle on the x-axis is if and that on the y-axis is if .

- General equation of second order degree in x and y represents a circle if and only if:
- coefficient of equals coefficient of , that is,
- coefficient of is zero, that is ,

**Section III: Some results regarding circles:**

*Position of a point with respect to a circle:*Point lies outside, on or inside a circle , according as*Parametric coordinates*of any point on the circle are given by with . In particular, parametric coordinates of any point on the circle.- An equation of the
*tangent*to the circle at the point on the circle is - An equation of the
*normal*to the circle at the point on the circle is - Equations of the
*tangent*and*normal*to the circle at the point on the circle are, respectively, and - The line is a tangent to the circle if and only if .
- The lines are tangents to the circle , for all finite values of m. If m is infinite, the tangents are .
- An equation of the
*chord*of the circle , whose mid-point is is , where and . In particular, an equation of the chord of the circle , whose mid-point is is . - An equation of the
*chord of contact*of the tangents drawn from a point outside the circle is .(S and T are as defined in (8) above). *Length of the tangent*drawn from a point outside the circle , to the circle, is . (S and ) are as defined in (8) above.)- Two circles with centres and and radii , respectively, (i)
*touch each other externally*if . the point of contact is and (ii)*touch each other internally*if , where ; the point of contact is - An equation of the
*family of circles*passing through the points and is , where

- An equation of the family of circles which touch the line at for any finite value of m is . If m is infinite, the equation becomes .
- Let QR be a chord of a circle passing through the point and let the tangents at the extremities Q and R of this chord intersect at the point . Then, locus of L is called the
of P with respect to the circle, and P is called the**polar****pole of its polar.***Equation of the polar of*with respect to the circle is , where T is defined as above.- If the polar of P with respect to a circle passes through Q, then the polar of Q with respect to the same circle passes through P. Two such points P and Q, are called
**conjugate points***of the same circle.*

- If lengths of the tangents drawn from a point P to the two circles and are equal, then the locus of P is called the
of the two circles and , and its equation is , that is,*radical axis*- Radical axis of two circles is perpendicular to the line joining their circles.
- Radical axes of three circles, taken in pairs, pass through a fixed point called the
of the three circles, if the centres of these circles are non-collinear.*radical centre*

**4: Special Forms of Equation of a Circle:**

- An equation of a circle with centre and radius is . This touches the co-ordinate axes at the points and .
- An equation of a circle with centre , radius is . This touches the x-axis at .
- An equation of a circle with centre and radius is . This circle passes through the origin , and has intercepts a and b on the x and y axes, respectively.

**5: Systems of Circles:**

Let ; and and .

- If two circles and intersect at real and distinct points, then where represents a family of circles passing through these points, where is a parameter, and when represents the chord of the circles.
- If two circles and touch each other, then represents equation of the common tangent to the two circles at their point of contact.
- If two circles and intersect each other
*orthogonally*(the tangents at the point of intersection of the two circles are at right angles), then . - If the circle intersects the line at two real and distinct points, then represents a family of circles passing through these points.
- If is a tangent to the circle at P, then represents a family of circles touching at P, and having as the common tangent at P.
*Coaxial Circles:*A system of circles is said to be coaxial if every pair of circles of the system have the same radical axis. The*simplest form*of the equation of a coaxial system of circles is : , where g is a variable and c is constant, the common radical axis of the system being y-axis and the line of centres being x-axis.*The Limiting points of the coaxial system of circles*are the members of the system which are of zero radius. Thus, the limiting points of the coaxial system of circles are if . The equation () represents a family of coaxial circles, two of whose members are given to be and .*Conjugate systems (or orthogonal systems) of circles :*Two system of circles such that every circle of one system cuts every circle of the other system orthogonally are said to be conjugate system of circles. For instance, and , where g and f are variables and c is constant, represent two systems of coaxial circles which are conjugate.

**6: Common tangents to two circles:**

If and are two circles with centres and and radii and respectively, then we have the following results regarding their common tangents:

- When , that is, distance between the centres is greater than the sum of their radii, the two circles do not intersect with each other, and four common tangents can be drawn to circles. Two of them are direct common tangents and other two are transverse common tangents. The points of intersection of direct common tangents and transverse common tangents respectively, always lie on the line joining the centres of the two circles and divide it externally and internally respectively in the ratio of their radii.
- When , that is, the distance between the centres is equal to the sum of their radii, the two circles touch each other externally, two direct tangents are real and distinct and the transverse tangents coincide.
- When , that is, the distance between the centres is less than the sum of the radii, the circles intersect at two real and distinct points, the two direct common tangents are real and distinct while the transverse common tangents are imaginary.
- When with , that is, the distance between the centres is equal to the difference of their radii, the circles touch each other internally, two direct common tangents are real and coincident, while the transeverse common tangents are imaginary.
- When , with , that is, the distance between the centres is less than the difference of the radii, one circle with smaller radius lies inside the other and the four common tangents are all imaginary.

*To be continued,*

*Nalin Pithwa.*

September 23, 2017 – 11:08 am

I have a collection of some “random”, yet what I call ‘beautiful” questions in Co-ordinate Geometry. I hope kids preparing for IITJEE Mains or KVPY or ISI Entrance Examination will also like them.

Problem 1:

Given n straight lines and a fixed point O, a straight line is drawn through O meeting lines in the points , , , , and on it a point R is taken such that

Show that the locus of R is a straight line.

Solution 1:

Let equations of the given lines be , , and the point O be the origin .

Then, the equation of the line through O can be written as where is the angle made by the line with the positive direction of x-axis and r is the distance of any point on the line from the origin O.

Let be the distances of the points from O which in turn and , where .

Then, coordinates of R are and of are where .

Since lies on , we can say for

, for

…as given…

Hence, the locus of R is which is a straight line.

Problem 2:

Determine all values of for which the point lies inside the triangle formed by the lines , , .

Solution 2:

Solving equations of the lines two at a time, we get the vertices of the given triangle as: , and .

So, AB is the line , AC is the line and BC is the line

Let be a point inside the triangle ABC. (please do draw it on a sheet of paper, if u want to understand this solution further.) Since A and P lie on the same side of the line , both and must have the same sign.

or which in turn which in turn either or ….call this relation I.

Again, since B and P lie on the same side of the line , and have the same sign.

and , that is, …call this relation II.

Lastly, since C and P lie on the same side of the line , we have and have the same sign.

that is

or ….call this relation III.

Now, relations I, II and III hold simultaneously if or .

Problem 3:

A variable straight line of slope 4 intersects the hyperbola at two points. Find the locus of the point which divides the line segment between these two points in the ratio .

Solution 3:

Let equation of the line be where c is a parameter. It intersects the hyperbola at two points, for which , that is, .

Let and be the roots of the equation. Then, and . If A and B are the points of intersection of the line and the hyperbola, then the coordinates of A are and that of B are .

Let be the point which divides AB in the ratio , then and , that is, …call this equation I.

and ….call this equation II.

Adding I and II, we get , that is,

….call this equation III.

Subtracting II from I, we get

so that the locus of is

More later,

Nalin Pithwa.

September 22, 2017 – 3:06 am

Problem 1:

The line joining and is produced to the point so that , then find the value of .

Solution 1:

As M divides AB externally in the ratio , we have and which in turn

Problem 2:

If the circumcentre of a triangle lies at the origin and the centroid in the middle point of the line joining the points and , then where does the orthocentre lie?

Solution 2:

From plane geometry, we know that the circumcentre, centroid and orthocentre of a triangle lie on a line. So, the orthocentre of the triangle lies on the line joining the circumcentre and the centroid , that is, , or . That is, the orthocentre lies on this line.

Problem 3:

If a, b, c are unequal and different from 1 such that the points , and are collinear, then which of the following option is true?

a:

b:

c:

d:

Solution 3:

Suppose the given points lie on the line then a, b, c are the roots of the equation :

, or

and , that is,

Eliminating l, m, n, we get

, that is, option (d) is the answer.

Problem 4:

If and are in A.P., with common difference a and b respectively, then on which line does the centre of mean position of the points with lie?

Solution 4:

Note: Centre of Mean Position is .

Let the coordinates of the centre of mean position of the points , be then

and

,

and

, and

, that is, the CM lies on this line.

Problem 5:

The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p and q are the intercepts of the line L on the new axes, then what is the value of ?

Solution 5:

Equation of the line L in the two coordinate systems is , and where are the new coordinate of a point when the axes are rotated through a fixed angle, keeping the origin fixed. As the length of the perpendicular from the origin has not changed.

or . So, the value is zero.

Problem 6:

Let O be the origin, and and are points such that and , then which of the following options is true:

a: P lies either inside the triangle OAB or in the third quadrant

b: P cannot lie inside the triangle OAB

c: P lies inside the triangle OAB

d: P lies in the first quadrant only.

Solution 6:

Since , P either lies in the first quadrant or in the third quadrant. The inequality represents all points below the line . So that and imply that either P lies inside the triangle OAB or in the third quadrant.

Problem 7:

An equation of a line through the point whose distance from the point has the greatest value is :

option i:

option ii:

option iii:

option iv:

Solution 7:

Let the equation of the line through be . If p denotes the length of the perpendicular from on this line, then

, say

then is greatest if and only if s is greatest.

Now,

so that . Also, , if , and

, if

and , if . So s is greatest for . And, thus, the equation of the required line is .

Problem 8:

The points , , Slatex C(4,0)$ and are the vertices of a :

option a: parallelogram

option b: rectangle

option c: rhombus

option d: square.

*Note: more than one option may be right. Please mark all that are right.*

Solution 8:

Mid-point of AC =

Mid-point of BD =

the diagonals AC and BD bisect each other.

ABCD is a parallelogram.

Next, and and since the diagonals are also equal, it is a rectangle.

As and , the adjacent sides are not equal and hence, it is neither a rhombus nor a square.

Problem 9:

Equations and will represent the same line if

option i:

option ii:

option iii:

option iv:

Solution 9:

The two lines will be identical if there exists some real number k, such that

, and , and .

or

or , and

or

That is, or , or .

Next, . Hence, , or .

Problem 10:

The circumcentre of a triangle with vertices , and lies at the origin, where and . Show that it’s orthocentre lies on the line

Solution 10:

As the circumcentre of the triangle is at the origin O, we have , where r is the radius of the circumcircle.

Hence,

Therefore, the coordinates of A are . Similarly, the coordinates of B are and those of C are . Thus, the coordinates of the centroid G of are

.

Now, if is the orthocentre of , then from geometry, the circumcentre, centroid, and the orthocentre of a triangle lie on a line, and the slope of OG equals the slope of OP.

Hence,

because .

Hence, the orthocentre lies on the line

.

Hope this gives an assorted flavour. More stuff later,

Nalin Pithwa.

September 19, 2017 – 1:56 pm

I. **Results regarding points in a plane:**

1a) **Distance Formula:**

The distance between two points and is given by . The distance from the origin to the point is .

1b) **Section Formula:**

If divides the join of and in the ratio with , then

, and

The positive sign is taken for internal division and the negative sign for external division. The *mid-point *of and is which corresponds to internal division, when . Note that for external division .

1c) **Centroid of a triangle:**

If is the centroid of the triangle with vertices , and then and

1d) **Incentre of a triangle:**

If is the incentre of the triangle with vertices , , and , then

, , a, b and c being the lengths of the sides BC, CA and AB, respectively of the triangle ABC.

1e) **Area of triangle:**

ABC with vertices , , and is ,

and is generally denoted by . Note that if one of the vertex is at , then .

*Note: *When A, B, and C are taken as vertices of a triangle, it is assumed that they are not collinear.

1f) **Condition of collinearity:**

Three points , , and are collinear if and only if

1g) **Slope of a line:**

Let and with be any two points. Then, the slope of the line joining A and B is defined as

where is the angle which the line makes with the positive direction of the x-axis, , except at . Which is possible only if and the line is parallel to the y-axis.

1h) **Condition for the points **, to form an equilateral triangle is

**II) Standard Forms of the Equation of a Line:**

- An equation of a line parallel to the x-axis is and that of the x-axis itself is .
- An equation of a line parallel to the y-axis is and that of the y-axis itself is .
- An equation of a line passing through the origin and (a) making an angle with the positive direction of the x-axis is , and (b) having a slope m is , and (c) passing through the point .
*Slope-intercept form:*An equation of a line with slope m and making an intercept c on the y-axis is .*Point-slope form:*An equation of a line with slope m and passing through is .*Two-point form:*An equation of a line passing through the points and is .*Intercept form:*An equation of a line making intercepts a and b on the x-axis and y-axis respectively, is .*Parametric form:*An equation of a line passing through a fixed point and making an angle with with with the positive direction of the x-axis is where r is the distance of any point on the line from the point . Note that and .*Normal form:*An equation of a line such that the length of the perpendicular from the origin on it is p and the angle which this perpendicular makes with the positive direction of the x-axis is , is .*General form:*In general, an equation of a straight line is of the form , where a, b, and c are real numbers and a and b cannot both be zero simultaneously. From this general form of the equation of the line, we can calculate the following: (i) the slope is (ii) the intercept on the x-axis is with and the intercept on the y-axis is with (iii) and and , the positive sign being taken if c is negative and vice-versa (iv) If denotes the length of the perpendicular from on this line, then and (v) the points and lie on the same side of the line if the expressions and have the same sign, and on the opposite side if they have the opposite signs.

**III) Some results for two or more lines:**

- Two lines given by the equations and are
*parallel*(that is, their slopes are equal) if*perpendicular*(that is, the product of their slopes is -1) if*identical*if*not parallel,*then- angle between them at their point of intersection is given by where being the slopes of the two lines.
- the coordinates of their points of intersection are
- An equation of any line through their point of intersection is where is a real number.

- An equation of a line parallel to the line is , and the distance between these lines is
- The three lines , and are
*concurrent*(intersect at a point) if and only if - Equations of the
*bisectors of the angles*between two intersecting lines and are . Any point on the bisectors is equidistant from the given lines. If is the angle between one of the bisectors and one of the lines such that , that is, , then that bisector bisects the acute angle between the two lines, that is, it is the*acute angle bisector*of the two lines. The other equation then represents the*obtuse angle**bisector*between the two lines. - Equations of the lines through and making an angle with the line , are where and where where is the slope of the given line. Note that and and when , .

**IV) Some Useful Points:**

To show that A, B, C, D are the vertices of a

**parallelogram:**show that the diagonals AC and BD bisect each other.**rhombus:**show that the diagonals AC and BD bisect each other and a pair of adjacent sides, say, AB and BC are equal.**square:**show that the diagonals AC and BD are equal and bisect each other, a pair of adjacent sides, say AB and BC are equal.**rectangle:**show that the diagonals AC and BD are equal and bisect each other.

**V) Locus of a point:**

To obtain the equation of a set of points satisfying some given condition(s) called locus, proceed as follows:

- Let be any point on the locus.
- Write the given condition involving h and k and simplify. If possible, draw a figure.
- Eliminate the unknowns, if any.
- Replace h by x and k by y and obtain an equation in terms of and the known quantities. This is the required locus.

**VI) Change of Axes:**

**Rotation of Axes:**if the axes are rotated through an angle in the anti-clockwise direction keeping the origin fixed, then the coordinates of a point with respect to the new system of coordinates are given by and .**Translation of Axes:**the shifting of origin of axes without rotation of axes is called*translation of axes.*If the origin is shifted to the point without rotation of the axes then the coordinates of a point with respect to the new system of coordinates are given by and .

*I hope to present some solved sample problems with solutions soon.*

Nalin Pithwa.