Question

A variable line passes through the fixed point ($\alpha,\,\beta$). The locus of the foot of the perpendicular from the origin on the line is,

• A. $x^{2}+y^{2}-\alpha x-\beta y = 0$
• B. $x^{2}-y^{2}+2\alpha x+2\beta y = 0$
• C. $\alpha x+\beta y \pm\sqrt{\left(\alpha^{2}+\beta^{2}\right)}= 0$
• D. $\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{\beta^{2}} = 1$

Answer 1

Answer: (A)

$x^{2}+y^{2}-\alpha x-\beta y = 0$

Answer 2

$\frac{k-\beta}{h-\alpha}\times\frac{k-0}{h-0} = -1$ $\Rightarrow$ locus is $y\left(y-\beta\right) = -x\left(x-\alpha\right)$ $\Rightarrow x^{2}+y^{2}-\alpha x-\beta y = 0$