Q.23. Circle, Parabola, In-radius

Question 23.

Consider the circle x^2 + y^2 = 9 and the parabola y^2 = 8x. They intersect at P and Q in the first and fourth quadrant 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. The radius of the incircle of the triangles PQR is_______?

Answer 23.
Solving the equations
x^2 + y^2 = 9 ... (1) and
y^2 = 8x ... (2),

x^2 + 8x - 9 = 0
=> (x - 1)(x + 9) = 0
=> x = 1 [x = -9 is not possible]
and y = ± 2√2
=> P(1, 2√2) and Q (1, - 2√2) are the points of intersection.

Eqn. of tangent at (1, 2√2) to the circle is
x + (2√2)y - 9 = 0
Its point of intersection with the x-axis is R(9, 0)

Area of A of ΔPQR = (1/2) modulus of determinant
l9   0   1 l
l1 2√2 1 l
l1 -2√2 1l
= 16√2.

Semiperimeter, s of ΔPQR
= (1/2) (PQ + QR + RS)
= (1/2) (4√2 + 6√2 + 6√2) [By distance formula]
= 8√2

Inradius, r = A/s = (16√2)/(8√2) = 2.


Link to YA!