Question

If the distance of two lines passing through origin from the point $(x_{1},y_{1})$ is $'d'$, then the equation of lines is

• A. $(xy_{1} - yx_{1})^{2} = d^{2}(x^{2} + y^{2})$
• B. $(x_{1}y_{1} - xy)^{2} = (x^{2} + y^{2})$
• C. $(xy_{1} + yx_{1})^{2} = (x^{2} - y^{2})$
• D. $(x^{2} - y^{2}) = 2(x_{1} + y_{1})$

Answer 1

Answer: (A)

$(xy_{1} - yx_{1})^{2} = d^{2}(x^{2} + y^{2})$

Answer 2

If the equation of line is $y = mx$ and the length of perpendicular drawn on it from the point $(x_{1},y_{1})$ is d, then $\frac{y_{1} - mx_{1}}{\sqrt{1 + m^{2}}} = \pm d \Rightarrow (y_{1} - mx_{1})^{2} = d^{2}(1 + m^{2}).$

But $m = \frac{y}{x},$ therefore on eliminating 'm' , the required equation is $(xy_{1} - yx_{1})^{2} = d^{2}(x^{2} + y^{2})$.