Question

If the tangent at the point $(h,k)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cuts the auxiliary circle in points whose ordinates are $y_1$ and $y_2$, then $\frac{1}{y_1}+\frac{1}{y_2}=$.

• A. $\frac{4}{k}$
• B. $\frac{3}{k}$
• C. $\frac{2}{k}$
• D. None of these

Answer 1

Answer: (C)

$\frac{2}{k}$

Equation of tangent of given hyperbola at point
$(h,k)$ is $\frac{hx}{a^2}-\frac{ky}{b^2}=1$ ...(i)
Equation of auxillary circle is $x^2+y^2=a^2$ .....(ii)
From (i) and (ii)
${\left[(1+\frac{ky}{b^2})\frac{a^2}{h}\right]}^2+y^2-a^2=0$
$\Rightarrow y^2(k^2{a}^4+b^4{h}^2)+2k{b}^2{a}^{4}y+b^4{a}^2(a^2-h^2)=0$
Now $\frac{y_1+y_2}{y_1{y}_2}=-\frac{2k{b}^2{a}^4}{b^4{a}^2(a^2-h^2)}=\frac{-2k{a}^2}{b^2{a}^2(1-\frac{h^2}{a^2})}$
$=\frac{-2k}{b^2\left(\frac{-k^2}{b^2}\right)}=\frac{2}{k}$