## Monthly Archives: December 2017

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

http://bookstore.ams.org/mbk-106/?_ga=2.200133421.246483269.1514729371-1691390747.1513651870

https://www.amazon.in/Dingo-Ate-Math-Book-Mathematics/dp/1470435217/ref=sr_1_1?s=books&ie=UTF8&qid=1514731044&sr=1-1&keywords=a+dingo+ate+my+math+book

comment: sheer joy…🙂 🙂 🙂

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

### Mathematics in the early Islamic World

https://martin56jones.wordpress.com/2017/12/27/maths-in-the-early-islamic-world-in-our-time-with-melvyn-bragg/

Thanks and regards to Melvyn Bragg from Nalin Pithwa.

### Xmas prize from Colleen Young :-)

https://colleenyoung.wordpress.com/2017/12/24/diffy/

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.?

—- From Lilavathi of 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 🙂 🙂 🙂

### 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) $x-y=0$

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) $x+2y=0$.

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) $x^{2}+ y^{2}-ay=0$.

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 :-) :-) :-)

http://www.crraoaimscs.org

Here’s to Statistics with love !

Nalin Pithwa.

### The Association of Mathematics Teachers of India

https://www.amtionline.com/book_list

(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!! 🙂 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 ! 🙂

Hats off to AMTI !!!

Nalin Pithwa.

### Childhood Maths — Gaurish Korpal’s blog

https://gaurish4math.wordpress.com/2017/12/07/childhood-maths-i/

https://gaurish4math.wordpress.com/2017/12/14/childhood-maths-ii/

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

Thanks Mr. Gaurish Korpal.

from Nalin Pithwa.