Question

A variable line passes through a fixed point $(x_1,y_1)$ and meets the axes at $A$ and $B$. If the rectangle $OAPB$ be completed, the locus of P is, ($O$ being the origin of the system of axes)

• A. $(y-y_1{)}^2=4(x-x_1)$
• B. $\frac{x_1}{x}+\frac{y_1}{y}=1$
• C. $x^2+y^2=x_1^2+y_1^2$
• D. $\frac{x^2}{2x_1^2}+\frac{y^2}{y_1^2}=1$

Answer 1

The correct option is B $\frac{x_1}{x}+\frac{y_1}{y}=1$

Let $P$ be $(h,k)$, then $A$ is $(h,0)$ and $B$ is $(0,k)$.
Equation of $AB$ is $\frac{x}{h}+\frac{y}{k}=1$.

It passes through $(x_1,y_1)$
$\Rightarrow \frac{x_1}{h}+\frac{y_1}{k}=1$
Hence, required locus is $\frac{x_1}{x}+\frac{y_1}{y}=1$