Circles
Circle tangent to line — centre and radius match
MJAT_TS7_P1
Grade 12
Question:
The line $y=2x$ touches a circle with centre $(0,\alpha)$, $\alpha>0$, radius $r$ at point $A_1$. $B_1$ is the diametrically opposite point. Given $\alpha+r=5+\sqrt{5}$. Match:
P)$\alpha$; Q)$r$; R)$A_1$; S)$B_1$
List-II: 1)$(-2,4)$; 2)$\sqrt{5}$; 3)$(-2,6)$; 4)$\sqrt{5}$; 5)$(2,4)$
A) P→4, Q→2, R→1, S→3
B) P→2, Q→4, R→1, S→3
C) P→4, Q→2, R→5, S→3
D) P→2, Q→4, R→3, S→5
Step-by-Step Solution
Key Concept: Distance from $(0,\alpha)$ to $y=2x$ (i.e., $2x-y=0$): $r=\alpha/\sqrt{5}$. With $\alpha+r=5+\sqrt{5}$: $\alpha(1+1/\sqrt{5})=5+\sqrt{5}=\sqrt{5}(\sqrt{5}+1)\Rightarrow\alpha=\sqrt{5}\cdot\frac{\sqrt{5}}{(1+1/\sqrt{5})}=5$. $r=5/\sqrt{5}=\sqrt{5}$.
Answer: **C**.
Correct Answer: C