A line passing through $P(4,2)$ cuts the coordinate axes at $A$ and $B$ respectively. If $O$ is the origin,…

1. $x^{-1}+y^{-1}=2$
2. $2 x^{-1}+y^{-1}=1$
3. $x^{-1}+2 y^{-1}=1$
4. $2 x^{-1}+3 y^{-1}=1$

Solution

Let a line cuts the coordinate axes at $A$ and $B$ respectively is

Now, coordinate of centre of the circumcircle of $\triangle O A B$ is mid-point of hypotenuse of right angle $\triangle O A B$ and it is mid-point of $A B$. So, centre of the circumcircle of $\triangle O A B$ is $\left(\frac{a}{2}, \frac{b}{2}\right)$. Now, from Eq. (ii), on taking locus of point $\left(\frac{a}{2}, \frac{b}{2}\right)$, we get $ \begin{aligned} \frac{2}{x}+\frac{1}{y} & =1 \\ \Rightarrow \quad 2 x^{-1}+y^{-1} & =1 \end{aligned} $ Hence, option (b) is correct

Asked in: AP EAMCET 2019 (20 Apr Shift 2)

Practice more Hyperbola questions on Aicharya