Question

A line is drawn through a fixed point (h, k) cutting the coordinate axes at P and Q respectively. The rectangle OPRQ is completed. Find the equation of locus of R.

• A. $\frac{x}{h}+\frac{y}{k}=1$
• B. $\frac{x}{y}+\frac{h}{k}=1$
• C. $\frac{h}{x}+\frac{k}{y}=1$
• D. $\frac{x}{k}+\frac{y}{k}=1$

Answer 1

Answer: (C)

$\frac{h}{x}+\frac{k}{y}=1$

Suppose the equation of line passing through point (h,k) is $y=mx+c\dots eqn\left(1\right)$

Point P is $(0,c)$ and point Q is $(\frac{-c}{m},0)$

Relation b/w c and m according to eqn (1) is $m=\frac{k-c}{h}$

For rectangle OPRS coordinates of R will be $(\frac{hc}{c-k},c)$

If coordinates of R are $(h_1,k_1)$ Then,

$h_1=\frac{hc}{c-k}\dots eqn\left(2\right)k_1=c\dots eqn\left(3\right)$

Putting the value of c from eqn(3) in eqn(2), We get

$h_1=\frac{h{k}_1}{k_1-k}\Rightarrow \frac{h}{h_1}+\frac{k}{k_1}=1$

substituting $(h_1,k_1)$ with (x,y), We get,

$\frac{h}{x}+\frac{k}{y}=1$