Question

The point on the hyperbola $3x^2-4y^2=72$ which is nearest to the line $3x+2y+1=0$ is

• A. $(-6,3)$
• B. (6, 3)
• C. $(-6,-3)$
• D. $(6,-3)$
• E. $(\sqrt{24},0)$

Answer 1

Answer: (A)

First we check which point satisfy the equation of hyperbola. All points in options are satisfied the equation of hyperbola $3x^2-4y^2=72$ . Now, we find one-by-one the length of perpendicular from point on Ellipse to the line
$3x+2y+1=0$ .
$p_{(-6,3)}=\frac{11}{\sqrt{13}}$
$p_{(6,3)}=\frac{25}{\sqrt{13}}$
$p_{(-6,-3)}=\frac{23}{\sqrt{13}}$
$p_{(6,-3)}=\frac{13}{\sqrt{13}}$ $p_{(\sqrt{24},0)}=\frac{3\sqrt{24}+1}{\sqrt{13}}$
The minimum length is $p_{(-6,3)}$ .
So, the point $(-6,3)$ is nearest to the given line.