Question

Equation of chord of contact of pair of tangents, drawn to ellipse $4x^2+9y^2=36$ from the point $m,n$, where $m\cdot n=m+n$, $m,n$ being non-zero positive integers, is

• A. $2x+9y=18$
• B. $2x+2y=1$
• C. $4x+9y=18$
• D. None of these

Answer 1

Answer: (C)

$4x+9y=18$

$4x^2+9y^2=36\frac{x^2}{9}+\frac{y^2}{4}=1m.n=m+nm.n-m=nm=\frac{n}{n-1}$

Both $m,n$ are positive integers so $n=1$ is not possible

For $n=2$ $,$ $m=\frac{2}{2-1}=2$

For $n=3$ $,m=\frac{3}{3-1}=\frac{3}{2}$

For $n=4$ $,m=\frac{4}{4-1}=\frac{4}{3}$

The numerator in the expression of $m$ is $1$ less than the denominator so one of them will be odd and other will be even, we will not get any integral value by dividing them

Except when $n=2$ So the point $(m,n)$ is $(2,2)$

Equation of chord of contact is $T=0$

$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1\frac{x\left(2\right)}{9}+\frac{y\left(2\right)}{4}=18x+18y=364x+9y=18$

So option $C$ is correct.