Question

If the line $5x+12y=9$ touches the hyperbola $x^2-9y^2=9$, then its point of contact is

• A. $(5,-\frac{4}{3})$
• B. $(5,\frac{4}{3})$
  
• C. $(-3,2)$
• D. $(3,0)$

Answer 1

The correct option is A $(5,-\frac{4}{3})$

Given line line $5x+12y=9$ or
$y=-\frac{5x}{12}+\frac{3}{4}$
any tangent to the given hyperbola is
$y=mx\pm \sqrt{9m^2-1}$
on comparing with given equation
$m=-\frac{5}{12},c=\frac{3}{4}$
Hence point of contact will be
$(-\frac{a^{2}m}{c},-\frac{b^2}{c})\equiv (5,-\frac{4}{3})$