Question
Draw the rough sketch and find the area of the region:
{(x, y) : y2 ≤ 8 x, x2 + x2 ≤ 9}
Solution
Given region is
{(x, y): y2 ≤ 8 x, x2 +y2 ≤ 9}
Consider the equations
y2 = 8 x ...(1)
and x2 + y2 = 9 ...(2)
From (1) and (2). we get,
x2 + 8 x = 9 or x2 + 8 x-9 = 0⇒ (x + 9)(x - 1) = 0
⇒ x = - 9, 1
which gives the abscissa of the points of intersection P and Q.
Rejecting negative value of x, we get, x = 1
Required area = Area of shaded region
= 2 (area of region APOA)
= 2[area of region AMPA + area of region MOPM]
