question_answer

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)         \[\frac{x}{k}+\frac{y}{h}=1\]

Correct Answer: C

Solution :

Let the coordinates of \[R(\alpha ,\beta ).\]Since, OPRQ is a rectangle, therefore coordinates of P and Q are \[(\alpha ,0)\] and \[(0,\beta )\] respectively. Now, equation of line PQ is                                 \[\frac{x}{\alpha }+\frac{y}{\beta }=1\] Since, \[(h,k)\]lies on this line \[\therefore \]  \[\frac{h}{\alpha }+\frac{k}{\beta }=1\] Hence, locus of \[R(\alpha ,\beta )\]is                                 \[\frac{h}{x}+\frac{k}{y}=1\]