-
Pages
-
Categories
- algebra
- applications of maths
- Basic Set Theory and Logic
- calculus
- careers in mathematics
- Cnennai Math Institute Entrance Exam
- co-ordinate geometry
- combinatorics or permutations and combinations
- Complex Numbers
- Fun with Mathematics
- geometry
- IITJEE Advanced
- IITJEE Advanced Mathematics
- IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha)
- IITJEE Foundation mathematics
- IITJEE Mains
- IMO International Mathematical Olympiad IMU
- Inequalities
- Information about IITJEE Examinations
- INMO
- ISI Kolkatta Entrance Exam
- KVPY
- mathematicians
- memory power concentration retention
- miscellaneous
- motivational stuff
- physicisrs
- Pre-RMO
- probability theory
- pure mathematics
- RMO
- RMO Number Theory
- Statistics
- time management
- Trigonometry
-
Archives
- April 2021
- March 2021
- January 2021
- November 2020
- October 2020
- September 2020
- August 2020
- July 2020
- June 2020
- May 2020
- April 2020
- March 2020
- February 2020
- January 2020
- December 2019
- November 2019
- October 2019
- September 2019
- August 2019
- June 2019
- May 2019
- March 2019
- February 2019
- January 2019
- November 2018
- October 2018
- September 2018
- August 2018
- July 2018
- June 2018
- May 2018
- April 2018
- March 2018
- February 2018
- January 2018
- December 2017
- November 2017
- October 2017
- September 2017
- August 2017
- July 2017
- June 2017
- May 2017
- April 2017
- March 2017
- February 2017
- January 2017
- November 2016
- October 2016
- September 2016
- August 2016
- July 2016
- June 2016
- May 2016
- April 2016
- March 2016
- February 2016
- January 2016
- December 2015
- November 2015
- October 2015
- September 2015
- August 2015
- July 2015
- June 2015
- May 2015
- April 2015
- March 2015
- February 2015
- January 2015
- December 2014
- November 2014
- October 2014
- September 2014
- August 2014
- July 2014
- June 2014
Monthly Archives: February 2018
Higher paying job than doctor/lawyer: some Singapore data
Some random problems/solutions in Coordinate Geometry II: IITJEE mathematics training
Question I:
Find the equation of the tangent to the circle at the point
. If the circle rolls up along this tangent by 2 units, find its equation in the new position.
Solution I:
The centre of the given circle is
and its radius is 2. Equation of the tangent at
to the circle is
or .
The slope of this line is showing that it makes an angle of 60 degrees with the x-axis. After the circle rolls up along the tangent at A through a distance 2 units, its centre moves from
to
. We now find the co-ordinates of
. Since
is parallel to the tangent at A and it passes through
,
, its equation is
, where
;
being at a distance 2 units on this line from
; its co-ordinates are
, that is,
.
Hence, the equation of the circle in the new position is
, which in turn implies that
.
Question 2:
A triangle has two of its sides along the axes, its third side touches the circle . Prove that the locus of the circumcentre of the triangle is
.
Solution 2:
The given circle has its centre at and its radius is a so that it touches both the axes along which lie the two sides of the triangle. Let the third side be
.
So that A is and B is
and the line AB touches the given circle. Since
is a right angle, AB is diameter of the circumcentre of the triangle AOB. So, the circumcentre
of the triangle AOB is the mid-point of AB,
that is, ,
.
Now, the equation of AB is , which touches the given circle,
.
Hence, the locus of is
.
Question 3:
A circle of radius 2 units rolls on the outerside of the circle , touching it externally. Find the locus of the centre of this outside circle. Also, find the equations of the common tangents of these two circles when the line joining the centres of the two circles make an angle of 60 degrees with x-axis.
Solution 3:
The centre C of the given circle is and its radius is 2. Let
be the centre of the outer circle touching the given circle externally then
, which in turn implies,
So, the locus of P is , or
.
Since the two circles touch each other externally,, there are 3 common tangents to these circles.
One will be perpendicular to the line joining the centres and the other two will be parallel to the line joining the centres as the radii of the two circles are equal, co-ordinates of P are given by
,
co-ordinates of M, the mid-point of CP is .
Hence, the equation of the common tangent perpendicular to CP is
.
Let the equation of the common tangent parallel to CP be .
Since it touches the given circle .
Hence, the other common tangents are .
Question 4:
If and
are the equations of two circles with radii r and
respectively, then show that the circles
cut orthogonally.
Solution 4:
Let the line of centres of the given circle be taken as the x-axis and its mid-point as the origin…Note this is the key simplifying assumption.
If the distance between the centres is 2a, the co-ordinates of the centre are and
. Hence, we get the following:
, that is,
and so that
, that is,
…call this I.
and and in turn
…call this II.
Now, since .
The circles I and II intersect orthogonally.
Question 5:
Let P, Q, R, S be the centres of the four circles each of which is cut by a fixed circle orthogonally. If ,
,
,
be the squares of the lengths of the tangents to the four circles from a point in their plane, then prove that
Solution 5:
Let the equations of the four circles be
,
, then centres of these circles are as follows:
,
,
, and
Let the fixed point in the plane be taken as the origin, then ,
,
and
. Let the equation of the fixed circle cutting the four circles orthogonally be
, then
, or
we get the following:
, for
.
Eliminating the unknowns g, f, c we get
or,
where ,
and ,
and
Hence, we get the following:
.
Homework Quiz Coordinate Geometry:
- OAB is any chord of a circle which passes through O, a point in the plane of the circle and meets it in points A and B. A point P is taken on this chord such that OP is (i) arithmetic mean (ii) geometric mean of OA and OB. Prove that the locus of P in either case is a circle. Determine the circle.
- Let
be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length OA.
- Let P, Q and R be the centres and
are the radii respectively of three coaxial circles. Show that
- If ABC be any triangle and
be the triangle formed by the polars of the points A, B, C with respect to a circle, so that
is the polar of A;
is the polar of B and
is the polar of C. Prove that the lines
,
and
meet in a point.
That’s all, folks !
Nalin Pithwa.
Listening math : Gaurish Korpal way :-) :-) :-)
https://gaurish4math.wordpress.com/2018/02/17/listening-maths/
Thanks Gaurish —
From — Nalin Pithwa.
PS: Hope my readers also read your blog regularly !! 🙂
Some random sample problems-solutions in Coordinate Geometry: I: IITJEE Mains Maths tutorials
Question I:
The point undergoes the following transformations, successively:
a) reflection about the line .
b) translation through a distance 2 units along the positive directions of the x-axis.
c) rotation through an angle of about the origin in the anticlockwise direction.
d) reflection about .
Hint: draw the diagrams at very step!
Ans:
Question 2:
are n points in a plane whose co-ordinates are
,
,
,
respectively.
,
is bisected at the point
,
is divided in the ratio 1:2 at
,
is divided in the ratio
at
,
is divided in the ratio
at
and so on until all n points are exhausted. Show that the co-ordinates of the final point so obtained are
.
Solution 2:
The co-ordinates of are
.
Now, divides
in the ratio
. Hence, the co-ordinates of
are
, or
.
Again, divides
in the ratio
. Therefore, the co-ordinates of
are
, or
.
Proceeding in this manner,we can show that the coordinates of the final point obtained will be
.
Remark: For a rigorous proof, prove the above by mathematical induction.
Question 3:
A line L intersects the three sides BC, CA, and AB of a triangle ABC at P, Q and R, respectively. Show that
Solution 3:
Let ,
, and
be the vertices of
, and let
be equation of the line L. If P divides BC in the ratio
, then the coordinates of P are
.
Also, as P lies on L, we have
…..call this relation I.
Similarly, we can obtain ….call this relation II.
and so, also, we can prove that …call this III.
Multiplying, I, II and III, we get the desired result.
The above is the famous Menelaus’s theorem of plane geometry proved with elementary tools of co-ordinate geometry. As a homework quiz, try proving the equally famous Ceva’s theorem of plane geometry with elementary tools of co-ordinate geometry.
Question 4:
A triangle has the lines and
as two of its sides, with
and
being roots of the equation
. If
is the orthocentre of the triangle, show that the equation of the third side is
.
Solution 4:
Since the given lines intersect at the origin, one of the triangle lies at the origin O(0,0). Let OA and OB be the given lines and
, respectively. Let the equation of AB be
. Now, as OH is perpendicular to AB, we have
,
, say…call this equation I
Also, the coordinates of A and B are respectively,
and
Therefore, the equation of AB is
or …call this II.
Similarly, the equation of BH is ….call this III.
Solving II and III, we get the coordinates of H. Subtracting III from II, we get
Since and
are the roots of the equation
, we have
and
.
because y=b for H.
.
Hence, the equation of AB is
More later,
Nalin Pithwa.
World Maths day : Mar 7 competitions
Thanks to Ms. Colleen Young:
https://colleenyoung.wordpress.com/2018/02/15/world-maths-day-2018/
Happy “e” day via Math: thanks to PlusMaths :-)
We owe a lot to the Indians who taught us counting — Albert Einstein
Jim Simons: another mathematical master and brightest billionaire !!
https://blogs.ams.org/mathgradblog/2018/02/03/incomplete-jim-simons-matters/#more-32329
Just sharing this AMS blog, one of the best motivational, living legends of Mathematics and its applications in minting money..
With lots of regards to Prof. Jim Simons, and AMS blogger, Jacob Gross — from Nalin Pithwa.
PS: I hope/wish such motivational examples exist in India also…It would be a great service to Indian mathematics..