Monthly Archives: December 2017

Math in Australia :-) A Dingo ate my Math Book !

AMS (American Mathematical Society) link:

Amazon India link:

comment: sheer joy…๐Ÿ™‚ ๐Ÿ™‚ ๐Ÿ™‚

— I am just waiting for the book to be available in Amazon India. ! —- Nalin Pithwa.

Motivation for Math from China with love :-)

Highly Motivational Math Video (in Chinese)

Mathematics in the early Islamic World

Thanks and regards to Melvyn Bragg from Nalin Pithwa.

Xmas prize from Colleen Young :-)

Merry Xmas and Happy Holidays!

Nalin Pithwa

Motivation for Combinatorics: “Lilavathi”

Say mathematician, how many are the combinations in one composition with ingredients of six different tastes — sweet, pungent, astringent, sour, salt and bitter — taking them by ones, twos, or threes, etc.?

—- Fromย Lilavathiย ofย Bhaskara (the great twelfth century mathematician of India).

— Nalin Pithwa.

PS: Almost all countries/nations have some culture of math just as they do of music and poetry and singing and dancing ๐Ÿ™‚ ๐Ÿ™‚ ๐Ÿ™‚ย 

Circles and System of Circles: IITJEE Mains: some solved problems I

Part I: Multiple Choice Questions:

Example 1:

Locus of the mid-points of the chords of the circle x^{2}+y^{2}=4 which subtend a right angle at the centre is (a) x+y=2 (b) x^{2}+y^{2}=1 (c) x^{2}+y^{2}=2 (d) x-y=0

Answer 1: C.

Solution 1:

Let O be the centre of the circle x^{2}+y^{2}=4, and let AB be any chord of this circle, so that \angle AOB=\pi /2. Let M(h,x) be the mid-point of AB. Then, OM is perpendicular to AB. Hence, (AB)^{2}=(OA)^{2}+(AM)^{2}=4-2=2 \Longrightarrow h^{2}+k^{2}=2. Therefore, the locus of (h,k) is x^{2}+y^{2}=2.

Example 2:

If the equation of one tangent to the circle with centre at (2,-1) from the origin is 3x+y=0, then the equation of the other tangent through the origin is (a) 3x-y=0 (b) x+3y=0 (c) x-3y=0 (d) x+2y=0.

Answer 2: C.

Solution 2:

Since 3x+y=0 touches the given circle, its radius equals the length of the perpendicular from the centre (2,-1) to the line 3x+y=0. That is,

r= |\frac{6-1}{\sqrt{9+1}}|=\frac{5}{\sqrt{10}}.

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

|\frac{2m+1}{\sqrt{1+m^{2}}}|=\frac{5}{\sqrt{10}}=25(1+m^{2})=10(2m+1)^{2} \Longrightarrow 3m^{2}+8m-3=0, 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 x-3y=0.

Example 3.

A variable chord is drawn through the origin to the circle x^{2}+y^{2}-2ax=0. The locus of the centre of the circle drawn on this chord as diameter is (a) x^{2}+y^{2}+ax=0 (b) x^{2}+y^{2}+ay=0 (c) x^{2}+y^{2}-ax=0 (d) x^{2}+ y^{2}-ay=0.

Answer c.

Solution 3:

Let (h,k) be the centre of the required circle. Then, (h,k) being the mid-point of the chord of the given circle, its equation is hx+ky-a(x+h)=h^{2}+k^{2}-2ah.

Since it passes through the origin, we have -ah=h^{2}+k^{2}-2ah \Longrightarrow h^{2}+k^{2}-ah=0.

Hence, locus of (h,k) is x^{2}+y^{2}-ax=0.

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) m(m+n) (b) m+n (c) n(m+n) (d) (1/2)(m+n).

To be continued,

Nalin Pithwa.

Announcement: Tenth Statistics Olympiad 2018 :-) :-) :-)

Here’s to Statistics with love !

Nalin Pithwa.


The Association of Mathematics Teachers of India

(I found this v nice organization and the list of its v cheap, high quality publications in Math for kids in a blog of Mr. Gaurish Korpal.)

Quite frankly, these mathematics teachers are doing/have done profound service to India’s budding, aspiring generations of child mathematicians!! ๐Ÿ™‚ And, also to many parents in India, who mostly (in my personal opinion) think of only law, engineering, and medicine as the only respectable professions…:-( like Professor “Virus” of the famous movie, Three Idiots ! ๐Ÿ™‚

Hats off to AMTI !!!

Nalin Pithwa.

Childhood Maths — Gaurish Korpal’s blog

Inspirational to all kids, and hopefully, educative to Indian parents also, if I may add…

Thanks Mr. Gaurish Korpal.

from Nalin Pithwa.