Question

The coordinates of a point on the hyperbola, $\frac{x^2}{24}-\frac{y^2}{18}=1,$ which is nearest to the line $3x+2y+1=0$ are

• A. $(6,3)$
• B. $(-6,-3)$
• C. $(6,-3)$
• D. $(-6,3)$

Answer 1

The correct option is D $(-6,3)$

Any tangent to the hyperbola is
$y=mx\pm \sqrt{24m^2-18}$
for line $y=-\frac{3}{2}x-\frac{1}{2}$ to have minimum distance from hyperbola tangent at the nearest point on the hyperbola should be parallel to given line
$\therefore m=-\frac{3}{2},c=\pm 6$
and point of contact will be
$(-\frac{a^{2}m}{c},-\frac{b^2}{c})\equiv (6,-3),(-6,3)$
$\because$ distance from the line $3x+2y+1=0$ is minimum
Hence required point is $(-6,3)$