## Monthly Archives: February 2018

### Some random problems/solutions in Coordinate Geometry II: IITJEE mathematics training

Question I:

Find the equation of the tangent to the circle $x^{2}-y^{2}-4x-8y+16=0$ at the point $(2+\sqrt{3},3)$. If the circle rolls up along this tangent by 2 units, find its equation in the new position.

Solution I:

The centre $C_{1}$ of the given circle is $(2,4)$ and its radius is 2. Equation of the tangent at $A(2+\sqrt{3},3)$ to the circle is

$x(2+\sqrt{3})+3y-2(x+2+\sqrt{3})-4(y+3)+16=0$

or $\sqrt{3}x-y-2\sqrt{3}=0$.

The slope of this line is $\sqrt{3}$ 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 $C_{1}$ to $C_{2}$. We now find the co-ordinates of $C_{2}$. Since $C_{1}C_{2}$ is parallel to the tangent at A and it passes through $C_{1}$$(2,4)$, its equation is $\frac{x-2}{\cos {\theta}} = \frac{y-4}{\sin {\theta}}$, where $\theta=60 \deg$; $C_{2}$ being at a distance 2 units on this line from $C_{1}$; its co-ordinates are

$(2\cos {\theta}+2, 2\sin{\theta}+4)$, that is, $(3, 4+\sqrt{3})$.

Hence, the equation of the circle in the new position is

$(x-3)^{2}+(y-(4+\sqrt{3})^{2})=2^{2}$, which in turn implies that

$x^{2}+y^{2}-6x-2(4+\sqrt{3})y+8(3+\sqrt{3})=0$.

Question 2:

A triangle has two of its sides along the axes, its third side touches the circle $x^{2}+y^{2}-2ax-2ay+a^{2}=0$. Prove that the locus of the circumcentre of the triangle is

$a^{2}-2a(x+y)+2xy=0$.

Solution 2:

The given circle has its centre at $C(a,a)$ 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 $\frac{x}{p} + \frac{y}{p}=1$.

So that A is $(p,a)$ and B is $(a,q)$ and the line AB touches the given circle. Since $\angle AOB$ is a right angle, AB is diameter of the circumcentre of the triangle AOB. So, the circumcentre $P(h,k)$ of the triangle AOB is the mid-point of AB,

that is, $2h=p$, $2k=q$.

Now, the equation of AB is $\frac{x}{p} + \frac{y}{q}=1$, which touches the given circle,

$\frac{a(p+q)-pq}{\sqrt{p^{2}+q^{2}}}=a$

$\Longrightarrow a^{2}(p+q)^{2}+p^{2}-2apq(p+q)=a^{2}(p^{2}+q^{2})$

$\Longrightarrow 2a^{2}-2a(p+q)+pq=0$

$2a^{2}-2a(2h+2k)+2h-2k=0$.

Hence, the locus of $P(h,k)$ is $a^{2}-2a(x+y)+2xy=0$.

Question 3:

A circle of radius 2 units rolls on the outerside of the circle $x^{2}+y^{2}+4x=0$, 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 $(-2,0)$ and its radius is 2. Let $P(h,k)$ be the centre of the outer circle touching the given circle externally then $CP=2+2=4$, which in turn implies,

$(h+2)^{2}+k^{2}=4^{2}$

So, the locus of P is $(x+2)^{2}+y^{2}=16$, or $x^{2}+y^{2}+4x-12=0$.

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

$\frac{h+2}{\cos{60 \deg}} = \frac{k-0}{\sin{60 \deg}}=4 \Longrightarrow h=0, k=2\sqrt{3}$,

co-ordinates of M, the mid-point of CP is $(-1,\sqrt{3})$.

Hence, the equation of the common tangent perpendicular to CP is

$y-\sqrt{3}=-\frac{1}{\sqrt{3}}(x+1) \Longrightarrow x+\sqrt{3}y-2=0$.

Let the equation of the common tangent parallel to CP be $\sqrt{3}x-y+\lambda=0$.

Since it touches the given circle $\frac{}{}= \pm 2 \Longrightarrow \lambda = 2\sqrt{3} \pm 4$.

Hence, the other common tangents are $\sqrt{3}x -y \pm 4 + 2\sqrt{3}=0$.

Question 4:

If $S=0$ and $S^{'}=0$ are the equations of two circles with radii r and $r^{'}$ respectively, then show that the circles $\frac{S}{r} \pm \frac{S^{'}}{r}=0$ 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 $(a,0)$ and $(-a,0)$. Hence, we get the following:

$S \equiv (x-a)^{2}+y^{2}-r^{2}=0$, that is,

$S^{'} \equiv x^{2}+y^{2}-2ax +a^{2}-r^{2}=0$

and $S^{'} \equiv x^{2}+y^{2}+2ax+a^{2}-(r^{'})^{2}=0$ so that $\frac{}{} + \frac{}{}=0 \Longrightarrow Sr^{'}+S^{'}r^{'}=0$, that is,

$\Longrightarrow (r+r^{'})(x^{2}+y^{2}+a^{2})-2ax(r^{'}-r)-rr^{'}(r+r^{'})=0$

$\Longrightarrow x^{2}+y^{2}-2a\frac{r^{'}-r}{r^{'}+r}x + a^{2}-rr^{'}=0$…call this I.

and $\frac{S}{r} - \frac{S^{'}}{r^{'}}=0$ and in turn $\Longrightarrow x^{2}+y^{2}-2a\frac{r^{'}+r}{r^{'}-r}+a^{2}+rr^{'}=0$…call this II.

Now, since $2(\{ -a\frac{r^{'}-r}{r^{'}+r}\})(\{-a\frac{r^{'}+r}{r^{'}-r} \})+ 2\times 0 \times 0 =2a^{2}=(a^{2}-rr^{'}) +(a^{2}+rr^{'})$.

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 $t_{1}^{2}$, $t_{2}^{2}$, $t_{3}^{2}$, $t_{4}^{2}$ be the squares of the lengths of the tangents to the four circles from a point in their plane, then prove that

$t_{1}^{2}\Delta QRS +t_{2}^{2}\Delta RSP + t_{3}^{2}\Delta SPQ + t_{4}^{2}\Delta PQR=0$

Solution 5:

Let the equations of the four circles be

$x^{2}+y^{2}+2g_{i}x+2f_{i}y+c_{i}=0$, $i=1,2,3,4$, then centres of these circles are as follows:

$P(-g_{1},-f_{1})$, $Q(-g_{2},-f_{2})$, $R(-g_{3},-f_{3})$, and $S(-g_{4},-f_{4})$

Let the fixed point in the plane be taken as the origin, then $t_{1}^{2}=c_{1}$, $t_{2}^{2}=c_{2}$, $t_{3}^{2}=c_{3}$ and $t_{4}^{2}=c_{4}$. Let the equation of the fixed circle cutting the four circles orthogonally be

$x^{2}+y^{2}+2gx + 2fy +c=0$, then $2gg_{1}+2ff_{1}=c+c_{1}=c+t_{1}^{2}$, or

we get the following:

$2gg_{i}+2ff_{i}-c-t_{i}^{2}=0$, for $i=1,2,3,4$.

Eliminating the unknowns g, f, c we get

$\left| \begin{array}{cccc} 2g_{1} & 2f_{1} & -1 & -t_{1}^{2} \\ 2g_{2} & 2f_{2} & -1 & -t_{2}^{2}\\ 2g_{3} & 2f_{3} & -1 & -t_{3}^{2}\\ 2g_{4} & 2f_{4} & -1 & -t_{4}^{2} \end{array} \right|$

or, $t_{1}^{2}|D_{1}|-t_{2}^{2}|D_{2}|+t_{3}^{2}|D_{3}|-t_{4}^{2}|D_{4}|=0$

where $|D_{1}|= \left| \begin{array}{ccc} g_{2} & f_{2} & 1\\ g_{1} & f_{1} & 1 \\ g_{4} & f_{4} & 1 \end{array} \right |=2\Delta QRS$,

and $|D_{2}|=2\Delta PRS$, $|D_{3}|=2\Delta PQS$ and $|D_{4}|=2\Delta PQR$

Hence, we get the following:

$t_{1}^{2}\Delta QRS + t_{2}^{2}\Delta RSP + t_{3}^{2}SPQ + t_{4}^{2}PQR=0$.

Homework Quiz Coordinate Geometry:

1. 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.
2. Let $2x^{2}+y^{2}-3xy=0$ 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.
3. Let P, Q and R be the centres and $r_{1}, r_{2}, r_{3}$ are the radii respectively of three coaxial circles. Show that $r_{1}^{2}QR + r_{2}^{2}RP + r_{3}^{2}PQ=-PQ. QR. RP$
4. If ABC be any triangle and $A^{'}B^{'}C^{'}$ be the triangle formed by the polars of the points A, B, C with respect to a circle, so that $B^{'}C^{'}$ is the polar of A; $C^{'}A^{'}$ is the polar of B and $A^{'}B^{'}$ is the polar of C. Prove that the lines $AA^{'}$, $BB^{'}$ and $CC^{'}$ 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 $(4,1)$ undergoes the following transformations, successively:

a) reflection about the line $y=x$.

b) translation through a distance 2 units along the positive directions of the x-axis.

c) rotation through an angle of $\pi/4$ about the origin in the anticlockwise direction.

d) reflection about $x=0$.

Hint: draw the diagrams at very step!

Ans: $(1/\sqrt{2}, 7/\sqrt{2})$

Question 2:

$A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ are n points in a plane whose co-ordinates are $(x_{1}, y_{1})$, $(x_{2},y_{2})$, $\ldots$, $(x_{n},y_{n})$ respectively. $A_{1}$, $A_{2}$ is bisected at the point $G_{1}$, $G_{1}A_{3}$ is divided in the ratio 1:2 at $G_{2}$, $G_{2}A_{4}$ is divided in the ratio $1:3$ at $G_{3}$, $G_{3}A_{3}$ is divided in the ratio $1:4$ at $G_{4}$ and so on until all n points are exhausted. Show that the co-ordinates of the final point so obtained are

$(\frac{1}{n}(x_{1}+x_{2}+ \ldots + x_{n}) , \frac{1}{n}(y_{1}+y_{2}+ \ldots + y_{n}) )$.

Solution 2:

The co-ordinates of $G_{1}$ are $(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2})$.

Now, $G_{2}$ divides $G_{1}A_{3}$ in the ratio $1:2$. Hence, the co-ordinates of $G_{2}$ are

$( \frac{1}{3}(\frac{2(x_{1}+x_{2})}{2}+x_{3}), \frac{1}{3}(\frac{3(y_{1}+y_{2})}{2}+y_{3}))$, or $(\frac{x_{1}+x_{2}+x_{3}}{3},\frac{y_{1}+y_{2}+y_{3}}{3})$.

Again, $G_{3}$ divides $G_{2}A_{4}$ in the ratio $1:4$. Therefore, the co-ordinates of $G_{3}$ are $(\frac{1}{4}(\frac{3(x_{1}+x_{2}+x_{3})}{3}+x_{4}) ,\frac{1}{4}(\frac{3(y_{1}+y_{2}+y_{3})}{3}+y_{4}) )$, or

$( \frac{x_{1}+x_{2}+x_{3}+x_{4}}{4},\frac{y_{1}+y_{2}+y_{3}+y_{4}}{4} )$.

Proceeding in this manner,we can show that the coordinates of the final point obtained will be

$(\frac{1}{n}(x_{1}+x_{2}+x_{3}+\ldots + x_{n}),\frac{1}{n}(y_{1}+y_{2}+y_{3}+\ldots + y_{n}))$.

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 $\frac{BP}{PC}.\frac{CQ}{QA}.\frac{AR}{RB}=-1$

Solution 3:

Let $A(x_{1},y_{1})$, $B(x_{2},y_{2})$, and $C(x_{3},y_{3})$ be the vertices of $\triangle ABC$, and let $lx+my+n=0$ be equation of the line L. If P divides BC in the ratio $\lambda:1$, then the coordinates of P are $(\frac{\lambda x_{3}+x_{2}}{\lambda + 1} ,\frac{\lambda y_{3}+y_{2}}{\lambda + 1})$.

Also, as P lies on L, we have $l(\frac{\lambda x_{3}+x_{2}}{\lambda + 1})+m(\frac{\lambda y_{3}+y_{2}}{\lambda + 1})+n=0$

$\Longrightarrow \frac{lx_{2}+my_{2}+n}{lx_{3}+my_{3}+n}=\lambda=\frac{BP}{PC}$…..call this relation I.

Similarly, we can obtain $\frac{CQ}{QA}=-\frac{lx_{3}+my_{3}+n}{lx_{1}+my_{1}+n}$….call this relation II.

and so, also, we can prove that $\frac{AR}{RB}=-\frac{lx_{1}+my_{1}+n}{lx_{2}+my_{2}+n}$…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 $y=m_{1}x$ and $y=m_{2}x$ as two of its sides, with $m_{1}$ and $m_{2}$ being roots of the equation $bx^{2}+2hx+a=0$. If $H(a,b)$ is the orthocentre of the triangle, show that the equation of the third side is $(a+b)(ax+by)=ab(a+b-2h)$.

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 $y=m_{1}x$ and $y=m_{2}x$, respectively. Let the equation of AB be $lx+my=1$. Now, as OH is perpendicular to AB, we have

$\frac{b}{a}=\frac{m}{l}$, $\Longrightarrow \frac{l}{a}=\frac{m}{b}=k$, say…call this equation I

Also, the coordinates of A and B are respectively,

$(\frac{1}{l+mm_{1}}, \frac{m_{1}}{l+mm_{1}})$ and $(\frac{1}{l+mm_{2}} , \frac{m_{2}}{l+mm_{2}})$

Therefore, the equation of AB is

$(y-\frac{m_{1}}{l+mm_{1}})=-\frac{1}{m_{2}}(x-\frac{1}{l+mm_{1}})$

or $x+m_{2}y=\frac{1+m_{1}m_{2}}{1+mm_{1}}$…call this II.

Similarly, the equation of BH is $x+m_{1}y=\frac{1+m_{1}m_{2}}{1+mm_{2}}$….call this III.

Solving II and III, we get the coordinates of H. Subtracting III from II, we get

$y=\frac{(1+m_{1}m_{2})m}{l^{2}+lm(m_{1}+m_{2})+m^{2}m_{1}m_{2}}$

Since $m_{1}$ and $m_{2}$ are the roots of the equation $bx^{2}+2hx+a=0$, we have $m_{1}+m_{2}=-\frac{2h}{b}$ and $m_{1}m_{2}=a/b$.

$\Longrightarrow y=\frac{(a+b)m}{bl^{2}-2hlm+am^{2}} \Longrightarrow \frac{m}{b}=\frac{bl^{2}-2hlm+am^{2}}{a+b}$ because y=b for H.

$\Longrightarrow k=\frac{k^{2}(ba^{2}-2hab+ab^{2})}{a+b} \Longrightarrow k=\frac{a+b}{ab(a-2h+b)}$.

Hence, the equation of AB is

$ax+by=\frac{1}{k}=\frac{ab(a+b-2h)}{a+b}$

$\Longrightarrow (a+b)(ax+by)=ab(a+b-2h)$

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/

### 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..

### Paul Erdos, Mathematics, Russia and USA:

It is true …universally, including India…

Hats off to the “Intellect of the Wise Mathematicians”, late, adorable professor of mathematics, Paul Erdos.

— humble tribute …from Nalin Pithwa.

Paul Erdos, most profound quote ever made by any mathematician