Consider the Parabola $y^{2} = 4 x$ . A circle C is drawn, centred at (3, 0) , touching the parabola at A, B (No part of circle, lies outside the parabola), then which of the following is / are true

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Area of the smaller region, enclosed by the circle C & the Parabola is $\frac{β}{3} - π$ square units, where

$β = 10$

If the point of intersection of tangents drawn to the Parabola at A and B meet at $Q (β, γ)$ where $β = 3$

Area of the smaller region, enclosed by the circle C & the Parabola is $\frac{β}{3} - 2 π$ square units, where $β = 20$

$P (α, 0)$ be a point on the axis of the Parabola such that
area of the triangle $P A B = 2$ (area of the triangle OAB), (where O -is origin) then $α$ can be 3

answer is A, C.

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Detailed Solution

Required circle equation is, ${(x - 3)}^{2} + y^{2} = 8$ , &A(1,2), B(1,-2)
Required point $P (α, 0)$ in $(3, 0)$
$2 (\frac{1}{2} 4 \times 1) = \frac{1}{2} \times 4 \times h \Rightarrow h = 2$
Required Area = $2 [\int_{0}^{1} 2 \sqrt{x} d x - \int_{3 - \sqrt{2}}^{1} \sqrt{8 - {(x - 3)}^{2}} d x]$
$= 2 [\frac{4}{3} - {[- 1 \times 2 + 4 (- \frac{π}{4})] - [- \sqrt{2} \times 0 + 4 (\frac{- π}{2})]}]$