1. Tardigrade
  2. Question
  3. Mathematics
  4. Two circles are S1 ≡(x+3)2+y2=9 S2 ≡(x-5)2+y2=16 with centres C1 C2 A direct common tangent is drawn from a point P on x-axis which touches S1 S2 at Q R, respectively. Find the ratio of area of triangle PQC 1 triangle PRC 2.

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. 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 $\triangle PQC _1 \& \triangle PRC _2$.

Conic Sections

Solution:

$\triangle PQC_1$, and $\triangle PRC _2$ are similar
image
$\therefore \frac{\text { Area of } \Delta PQC _1}{\text { Area of } \Delta PRC _2}=\frac{\frac{1}{2} \times PQ \times r _1}{\frac{1}{2} \times PR \times r _2}$
$=\frac{ PQ }{ PR } \cdot \frac{r_1}{r_2}=\frac{r_1}{r_2} \cdot \frac{r_1}{r_2}=\frac{r_1^2}{r_2^2}=\frac{9}{16}$