Question

If $m_1$ and $m_2$ are the slopes of the tangents to the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ which pass through the point $(6,2)$, then find the value of $11m_1{m}_2$ and $11(m_1+m_2)$.

• A. $20$ & $24$
• B. $24$ & $20$
• C. $10$ & $12$
• D. $12$ & $10$

Answer 1

Answer: (A)

$20$ & $24$

Equation of tangent passing through $(6,2)$ is given by,
$(y-2)=m(x-6)\Rightarrow y=mx+(2-6m)$
Now for this equation to be tangent of given hyperbola, $c^2=a^2{m}^2-b^2$
$\Rightarrow (2-6m{)}^2=25m^2-16$
$\Rightarrow 11m^2-24m+20=0$
$\therefore m_1+m_2=\frac{24}{11},m_1{m}_2=\frac{20}{11}$
$\Rightarrow 11(m_1+m_2)=24,11m_1{m}_2=20$