JEE PYQ: Circle Question 1
Question 1 - 2021 (16 Mar Shift 1)
Let ABCD be a square of side of unit length. Let a circle $C_1$ centered at $A$ with unit radius is drawn. Another circle $C_2$ which touches $C_1$ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point $C$ to the circle $C_2$ meet the side AB at E. If the length of EB is $\alpha + \sqrt{3}\beta$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ is equal to ______.
Show Answer
Answer: (1)
Solution
Circle $C_1$: center $A(0,0)$, radius 1. $C_2$ touches $C_1$ and lines $AD$, $AB$: center $(r,r)$, radius $r$, with $(\sqrt{2}+1)r = 1$, so $r = \sqrt{2}-1$. Tangent from $C(1,1)$ to $C_2$ meets AB at E. By calculation, $EB = 2 - \sqrt{3}$, so $\alpha = 2$, $\beta = -1$, $\alpha + \beta = 1$.
BETA
