Question
A variable line through the point $$\left( {a,\,b} \right)$$ cuts the axes of reference at $$A$$ and $$B$$ respectively. The lines through $$A$$ and $$B$$ parallel to the $$y$$-axis and the $$x$$-axis respectively meet at $$P.$$ Then the locus of $$P$$ has the equation :
A.
$$\frac{x}{a} + \frac{y}{b} = 1$$
B.
$$\frac{x}{b} + \frac{y}{a} = 1$$
C.
$$\frac{a}{x} + \frac{b}{y} = 1$$
D.
$$\frac{b}{x} + \frac{a}{y} = 1$$
Answer :
$$\frac{a}{x} + \frac{b}{y} = 1$$
Solution :
$$\eqalign{ & {\text{Clearly,}}\,\,\frac{b}{{{x_1} - a}} = \tan \,\theta = \frac{{{y_1} - b}}{a} \cr & \therefore \,\,\,ab = {x_1}{y_1} - b{x_1} - a{y_1} + ab \cr & \Rightarrow \frac{a}{{{x_1}}} + \frac{b}{{{y_1}}} = 1 \cr} $$
$$\eqalign{ & {\text{Clearly,}}\,\,\frac{b}{{{x_1} - a}} = \tan \,\theta = \frac{{{y_1} - b}}{a} \cr & \therefore \,\,\,ab = {x_1}{y_1} - b{x_1} - a{y_1} + ab \cr & \Rightarrow \frac{a}{{{x_1}}} + \frac{b}{{{y_1}}} = 1 \cr} $$