Question

From the point $A(4,3)$ tangents are drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ to touch the ellipse at $B$ and $C$. EF is a tangent to the ellipse parallel to the line $BC$ and towards the point $A$. The distance of $A$ from EF is equal to

• A. $\frac{4\sqrt{18}-12}{5}$
• B. $\frac{24-4\sqrt{18}}{5}$
• C. $\frac{12-2\sqrt{18}}{5}$
• D. $\frac{12+2\sqrt{18}}{5}$

Answer 1

Answer: (B)

Equation of chord of contact at $A(4,3)$
$\frac{x}{4}+\frac{y}{3}=1$
Slope of line EF is $\frac{-3}{4}$
Equation of $EF$, (EF is tangent of ellipse)
$y=mx+\sqrt{a^2{m}^2+b^2}$
$y=\frac{-3}{4}x+\sqrt{16\cdot \frac{9}{16}+9}$
$y=\frac{-3}{4}x+\sqrt{18}$
$\text{ EF, }3x+4y-4\sqrt{18}=0$
$d=\left|\frac{12+12-4\sqrt{18}}{5}\right|=\left|\frac{24-4\sqrt{18}}{5}\right|$