Properties of Triangles 3 Question 5
5. In a $\triangle A B C$, let $\angle C=\pi / 2$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then $2(r+R)$ is equal to
$(2000,2 M)$
(a) $a+b$
(b) $b+c$
(c) $c+a$
(d) $a+b+c$
Passage Based Problems
Consider the circle $x^{2}+y^{2}=9$ and the parabola $y^{2}=8 x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $X$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $X$-axis at $S$.
$(2007,8 M)$
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Answer:
Correct Answer: 5. (a)
Solution:
- Here, $R^{2}=M C^{2}=\frac{1}{4}\left(a^{2}+b^{2}\right) \quad$ [by distance from origin] $=\frac{1}{4} c^{2} \quad$ [by Pythagoras theorem]
$$ \Rightarrow \quad R=\frac{c}{2} $$
Next, $\quad r=(s-c) \tan (C / 2)=(s-c) \tan \pi / 4=s-c$
$\therefore \quad 2(r+R)=2 r+2 R=2 s-2 c+c$
$$ \begin{aligned} & =a+b+c-c \\ & =a+b \end{aligned} $$
