A circle C_1 passes through the origin [O] and has diameter 4 on the positive x-axis.

Question

A circle $C_{1}$ passes through the origin $$\mathrm{O}$$ and has diameter 4 on the positive $x$-axis. The line $y=2 x$ gives a chord $$\mathrm{OA}$$ of circle $$\mathrm{C}_{1}$$. Let $$\mathrm{C}_{2}$$ be the circle with $$\mathrm{OA}$$ as a diameter. If the tangent to $$\mathrm{C}_{2}$$ at the point $$\mathrm{A}$$ meets the $x$-axis at $$\mathrm{P}$$ and $y$-axis at $$\mathrm{Q}$$, then $$\mathrm{QA}: \mathrm{AP}$$ is equal to :

Accepted Answer

Equation of C1

${x^2} + {y^2} - 4x = 0$

Intersection with

$y = 2x$

${x^2} + 4{x^2} - 4x = 0$

$5{x^2} - 4x = 0$

$$ \Rightarrow x = 0,{4 \over 5}$$

$$y = 0,{8 \over 5}$$

$$A:\left( {{4 \over 5},{8 \over 5}} \right)$$

Tangent of C2 at $$A\left( {{4 \over 5},{8 \over 5}} \right)$$

$$x + 2y = 4 \Rightarrow P:(4,0),\,Q:(0,2)$$

$QA:AP = 1:4$

Solutions · 1