Straight Lines Question 7
Question 7 - 29 January - Shift 2
A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^{2}=2 x$ and $x^{2}+y^{2}=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r^{2}$ is equal to
Show Answer
Answer: (10)
Solution:
Formula: Distance Formula,Area of Triangle
$S_1: y^{2}=2 x$
$S_2: x^{2}+y^{2}=4 x$
$P(2,2)$ is common point on $S_1 ; and ; S_2$
$T_1$ is tangent to $S_1$ at $P \quad \Rightarrow T_1: y \cdot 2=x+2$
$ \Rightarrow T_1: x-2 y+2=0 $
$T_2$ is tangent to $S_2$ at $P \quad \Rightarrow T_2: x .2+y \cdot 2=2(x+2)$
$ \Rightarrow T_2: y=2 $
& $L_3: x+y+2=0$ is third line
$PQ=a=\sqrt{20}$
$QR=b=\sqrt{8}$
$RP=c=6$
Area $(\triangle PQR)=\Delta=\frac{1}{2} \times 6 \times 2=6$
$\therefore r=\frac{abc}{4 \Delta}=\frac{\sqrt{160}}{4}=\sqrt{10} \Rightarrow r^{2}=10$
