Question

If the equation of the chord of contact of tangents drawn from a point $(a,b)$ to an ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ is given by $3x+4y=6.$ Then, find the point $(a,b).$

• A. $(8,6)$
• B. $(5,6)$
• C. $(2,5)$
• D. $(4,3)$

Answer 1

The correct option is A $(8,6)$

$\text{Given}:\frac{x^2}{16}+\frac{y^2}{9}=1$
and equation of chord of contact drawn from point $(a,b)$ is,
$3x+4y=6\cdots \cdots \left(1\right)$

We know, equation of chord of contact of tangents to the ellipse
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ drawn from $P(h,k)$ is $\frac{hx}{a^2}+\frac{ky}{b^2}=1$
$\therefore$ Equation of chord drawn from $(a,b)\text{is},$
$\frac{ax}{16}+\frac{by}{9}=1$
$\Rightarrow x\left(\frac{a}{16}\right)+y\left(\frac{b}{9}\right)=1\cdots \cdots \left(2\right)$
Eqn $\left(1\right)\text{and}\left(2\right)$ represent the same chord
$\therefore \frac{3}{\frac{a}{16}}=\frac{4}{\frac{b}{9}}=\frac{6}{1}$
$\Rightarrow \frac{3}{\frac{a}{16}}=6\text{and}\frac{4}{\frac{b}{9}}=6$
$a=8\text{and}b=6$
Hence, Required point is $(8,6).$