Question

The equation of tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ , which are perpendicular to the line $3x+4y=7$ , are

• A. $4x-3y=\pm \sqrt{20}$
• B. $4x-3y=\pm \sqrt{12}$
• C. $4x-3y=\pm \sqrt{2}$
• D. $4x-3y=\pm 1$

Answer 1

Answer: (A)

Let the equation of tangent, which is perpendicular to the line
$3x+4y=7$ , is $4x-3y=\lambda$ .
Since, it is a tangent to the ellipse.
$\therefore$ ${\lambda }^2=a^2{m}^2+b^2$
Here, $a^2=9,b^2=4$ and $m=\frac{4}{3}$
$\therefore$ ${\lambda }^2=9\times {\left(\frac{4}{3}\right)}^2+4$
$=16+4$
$\Rightarrow$ ${\lambda }^2=20$
$\Rightarrow$ $\lambda =\pm \sqrt{20}$
$\therefore$ Equation is $4x-3y=\pm \sqrt{20}$