The radius of the incircle of the triangle PQR is, Tangents to the circle at P and Q intersect the -axis at R and tangents to the parabola at P and Q intersect the -axis at S. [2007]

Question

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]

Q. The radius of the incircle of the triangle PQR is

Smart Achievers · Accepted Answer

(4)

$Radius of incircle$

$r = \frac{Δ}{s} = \frac{area (∆ P Q R)}{semi-perimeter of ∆ P Q R}$

$Here a = P R = 6 \sqrt{2}, b = Q P = 6 \sqrt{2}, c = P Q = 4 \sqrt{2}$

$and area (∆ P Q R) = \frac{1}{2} \times P Q \times T R = 16 \sqrt{2}$

$∴$ $Perimeter of ∆ P Q R = \frac{6 \sqrt{2} + 6 \sqrt{2} + 4 \sqrt{2}}{2} = 8 \sqrt{2}$

$∴$ $r = \frac{16 \sqrt{2}}{8 \sqrt{2}} = 2$

Solution

1 $4$

2 $3$

3 $\frac{8}{3}$

4 $2$

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