Question

The equation of the tangents to the ellipse $3x^2+4y^2=12,$ which are perpendicular to the line $y+2x=4,$ are

• A. $2y=x\pm 4$
• B. $y=2x\pm 2$
• C. $2y=-x\pm 4$
• D. $y=-2x\pm 2$

Answer 1

The correct option is A $2y=x\pm 4$

Slope of the given line $y+2x=4$ is $-2$
Given ellipse is $\frac{x^2}{4}+\frac{y^2}{3}=1$
Equation of tangent whose slope is $m$ is $y=mx\pm \sqrt{4m^2+3}\cdots \left(1\right)$

Given that tangent and given line are perpendicular to each other. So,
$m\times (-2)=-1$
$\Rightarrow m=\frac{1}{2}$
Now, from equation $\left(1\right)$
$y=\frac{1}{2}x\pm \sqrt{1+3}$
$\Rightarrow 2y=x\pm 4$