https://www.maa.org/programs/students/fun-math
cheers to MAA! 🙂
Nalin Pithwa.
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:
Section III: Some results regarding circles:
4: Special Forms of Equation of a Circle:
5: Systems of Circles:
Let ; and
and
.
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:
To be continued,
Nalin Pithwa.
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.
Kids now-a-days need counselling for a choice of career. In my humble opinion, excellence in any field of knowledge/human endeavour gives deep satisfaction as well as a means of livelihood. But, I am a mere mortal, most of my students are bright, ambitious, multi-talented and hard-working. I would like to present to them the views of one of my “idols”, though not a mathematician…
Dr. Shawna Pandya:
Hats off to Dr. Shawna Pandya, belated though from me…:-)
— Nalin Pithwa.
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.
Mathematicians are like Frenchmen: whatever you tell to them they translate in their own language and forthwith it is something entirely different. — GOETHE.
🙂 🙂 🙂
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:
III) Some results for two or more lines:
IV) Some Useful Points:
To show that A, B, C, D are the vertices of a
V) Locus of a point:
To obtain the equation of a set of points satisfying some given condition(s) called locus, proceed as follows:
VI) Change of Axes:
I hope to present some solved sample problems with solutions soon.
Nalin Pithwa.
Aber das ist English ! 🙂
Question 1:
Find the equation to that tangent to the parabola which is parallel to the straight line
. Find also its point of contact.
Question 2:
If P, Q and R are three points on the parabola whose coordinates are in geometric progression, prove that the tangents at P and R meet on the ordinate of Q.
Question 3:
is a double ordinate of the parabola
. Prove that the locus of the point of intersection of the normal at P and straight line through
parallel to the axis is the parabola
.
Question 4:
Show that in a parabola, the length of the focal chord varies inversely as the square of the distance of the vertex of the parabola from the focal chord.
Question 5:
Prove that the equation represents a parabola whose axis is parallel to the x-axis.. Find its vertex.
Question 6:
If the line touches the parabola
, find the length of the latus rectum.
Question 7a:
Prove that the circle described on any focal chord of a parabola as the diameter touches the directrix of the parabola.
Question 7b:
Show that the locus of the point, such that two of the normals drawn from it to the parabola coincide is
.
Question 7c:
If the normals at three points A, B and C on the parabola pass through the point
and cut the axis of the parabola in P, Q and R so that OP, OQ, OR are in AP, O being the vertex of the parabola, prove that the locus of the point S is
.
Question 8:
If P, Q and R be three conormal points on the parabola , the normals at which pass through the point T, and S is the focus of the parabola, then prove that
Question 9:
The tangents at P and Q to the parabola meet in T and the corresponding normals meet in R. If the locus of T is a straight line parallel to the axis of the parabola, prove that the locus of R is a straight line normal to the parabola.
Question 10a:
A variable chord PQ of the parabola is drawn parallel to the line
. If the parameters of the points P and Q on the parabola be
and
, then
. Also, show that the locus of the point of intersection of the normals at P and Q is
, which is itself a normal to the parabola.
Question 10b:
PQ is a chord of the parabola whose right bisector meets the axis in M and the ordinate of the mid-point of PQ meets the axis in L. Show that LM is constant and find LM.
Question 11:
Prove that the locus of a point P such that the slopes ,
,
of the three normals drawn to the parabola
from P be connected by the relation
is
.
Question 12a:
If the perpendicular drawn from P on the polar of P with respect to the parabola, , prove that the locus of P is the straight line
.
Question 12b:
Prove that the locus of the poles of the tangent to the parabola w.r.t. the circle
is
.
Question 13:
Tangents are drawn to the parabola at the points P and Q whose inclination to the axis are
,
. If A be the vertex of the parabola and the circles on AP and AQ as diameters intersect in R and AR be inclined at an angle
to the axis, then prove that
.
Question 14:
Through the vertex O of a parabola chords OP and OQ are drawn at right angles to one another. Prove that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also, find the locus of the middle point of PQ.
Question 15a:
Prove that the area of a triangle inscribed in a parabola is twice the area of the triangle formed by the tangents at the vertices of the triangle.
Question 15b:
Normals are drawn to the parabola at points A, B, and C whose parameter are
,
,
respectively. If these normals enclose a triangle PQR, then prove that its area is
. Also, prove that
.
Question 16:
Find the locus of the middle points of the chords of the ellipse which are drawn through the positive end of the minor axis.
Question 17:
Prove that the sum of the squares of the perpendiculars on any tangent to the ellipse from the points on the minor axis, each at a distance
from the centre, is
.
Question 18:
If and
are two variable points on the ellipse
such that
(some constant), then prove that the tangent at
is parallel to PQ.
Question 19:
P is a variable point on the ellipse whose foci are the points
, and the eccentricity is e. Prove that the locus of incentre of
is an ellipse whose eccentricity is
.
Question 20:
Consider the family of circles ,
. If in the first quadrant, the common tangent to a circle of this family and the ellipse
meets the coordinate axes at A and B, then find the equation of the locus of the mid-point of AB.
Question 21:
Let P be a point on the ellipse ,
. Let the line parallel to y-axis passing through P meet the circle
at the point Q such that P and Q are on the same side of the x-axis. For two positive real numbers, r and s, find the locus of the point R on PQ such that
as P varies over the ellipse.
Question 22:
If the eccentric angles of points P and Q on the ellipse be and
and
be the angle between the normals at P and Q, then prove that the eccentricity e is given by
.
Question 23:
A series of hyperbolas are such that the length of their transverse axis is 2a. Prove that the locus of a point P on each, such that its distance from transverse axis is equal to its distance from an asymptote is the curve:
.
Question 24:
A variable 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 1:2.
Question 25:
If the tangent at the point on the hyperbola
cuts the auxillary circle in points whose coordinates are
and
, then show that q is harmonic mean of
and
.
Question 26:
Show that the locus of poles with respect to the parabola of the tangents to the hyperbola
to the ellipse
.
Question 27:
The point P on the hyperbola with focus S is such that the tangent at P, the latus rectum through S and one asymptote are concurrent. Prove that SP is parallel to other asymptote.
Question 28:
If a triangle is inscribed in a rectangular hyperbola, prove that the orthocentre of the triangle lies on the curve.
Question 29:
A series of chords of the hyperbola touch the circle on the line joining the foci as diameter. Show that the locus of the poles of these chords with respect to the hyperbola is
.
Question 30:
Prove that the chord of the hyperbola which touches the conjugate hyperbola is bisected at the point of contact.
Cheers,
Nalin Pithwa.
Question 1:
Find the locus of the point of intersection of tangents to the ellipse , which are at right angles.
Solution 1:
Any tangent to the ellipse is …call this equation I.
Equation of the tangent perpendicular to this tangent is …call this Equation II.
The locus of the point of intersection of I and II is obtained by eliminating m between these equations. Squaring and adding we get:
which is a circle with its centre at the centre of the ellipse and radius equal to the length of the line joining the ends of the major and minor axis. This circle is called the director circle of the ellipse. 🙂
Question 2:
A tangent to the ellipse meets the ellipse
at P and Q. Prove that the tangents at P and Q of the ellipse
are at right angles.
Solution 2:
Let the tangent at to the ellipse
(equation I) meet the ellipse
(equation II) at P and Q.
Let the tangents at P and Q to II intersect at the point . Then, PQ is the chord of contact of the point
with respect to II and so its equation is
( Equation III).
PQ is also the tangent at to I and so its equation can be written as
(Equation IV)
Comparing III and IV, we get
,
,
,
the locus of is
or
, which is the director circle of the ellipse II. Hence, the tangents at P and Q to the ellipse II are at right angles (using the previous example). 🙂
Question 3:
Let d be the perpendicular distance from the centre of the ellipse to the tangent drawn at a point P on the ellipse. If
and
are the two foci of the ellipse, then prove that
.
Solution 3:
Equation of the tangent at the point on the given ellipse is
. Thus,
.
We know that
…call this equation I.
Also, ,
which in turn equals
,
which in turn equals
,
which in turn equals
,
which in turn equals
Now, from I, we get
Also,
which in turn equals
.
Hence, . 🙂
More later,
Nalin Pithwa.