Question

If m1 and m2 are the slopes of the tangents to the hyperbola x square by 25 minus y square by 16 is equal to 1 which pass through the point 6 comma two then 11 m1 m2 by 10 is equal to

Answer 1

Answer:

Equation of the hyperbola is

$\frac{x^2}{25}-\frac{y^2}{16}=1$

Equation of two lines passing through , (6,2), having slopes $m_{1}, {\text{and}} m_{2}$ and which are tangents to the hyperbola is

$\frac{y-6}{x-2}=m_{1}\\\\\frac{y-6}{x-2}=m_{2}$

$m_{1}x-2 m_{1}=y-6\\\\ m_{1}x-y=2 m_{1}-6\\\\ \frac{m_{1}x}{2m_{1}-6}-\frac{y}{2m_{1}-6}=1\\\\{\text{Similarly}}, \frac{m_{2}x}{2m_{2}-6}-\frac{y}{2m_{2}-6}=1$

Equation of tangent to the hyperbola through any point (a,b) is ,

$\frac{xa}{25}-\frac{yb}{16}=1$

Comparing the above equation ,with the equation of two tangents to get value of  $m_{1}, {\text{and}} m_{2}$.

$2m_{1}-6=25\\\\m_{1}=\frac{31}{2}\\\\2m_{2}-6=16\\\\m_{2}=11$

$\frac{11\times m_{1}\times m_{2}}{10}=11 \times \frac{31}{2} \times 11 \times \frac{1}{10}=\frac{3751}{20}=187.55$