Question

Two circles are
$S_1\equiv (x+3{)}^2+y^2=9$
$S_2\equiv (x-5{)}^2+y^2=16$
with centres $C_1\&C_2$
A direct common tangent is drawn from a point $P$ on $x$-axis which touches $S_1$ & $S_2$ at $Q\&R$, respectively. Find the ratio of area of $△PQ{C}_1\&△PR{C}_2$.

• A. $3:4$
• B. $9:16$
• C. $16:9$
• D. $4:3$

Answer 1

Answer: (B)

$△PQ{C}_1$, and $△PR{C}_2$ are similar

$\therefore \frac{\text{ Area of }\Delta PQ{C}_1}{\text{ Area of }\Delta PR{C}_2}=\frac{\frac{1}{2}\times PQ\times r_1}{\frac{1}{2}\times PR\times r_2}$
$=\frac{PQ}{PR}\cdot \frac{r_1}{r_2}=\frac{r_1}{r_2}\cdot \frac{r_1}{r_2}=\frac{r_1^2}{r_2^2}=\frac{9}{16}$