Verified Solution • 2025–26 Session moderate

QJEEAdv21Madd63: JEE Advanced 2021 Mathematics - Class 11 JEEAdv21Madd63 Solution: Consider the region R = {(x, y) R R : x 0 and y 2 4 x}. Let F…

Consider the region R = {(x, y) ∈\in R ×\times R : x ≥\ge 0 and y2 ≤\le 4 −- x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let ( α\alpha , β\beta ) be a point where the circle C meets the curve y2 = 4 −- x.

The value of α\alpha is ___________.

Step-by-Step Verified Solution

Given, x ≥\ge 0, y2 ≤\le 4 −- x

Let equation of circle be

(x −- h)2 + y2 = h2 .... (i)

JEE Advanced 2021 Paper 2 Online Mathematics - Circle Question 8 English Explanation
Solving Eq. (i) with y2 = 4 −- x, we get

x2 −- 2hx + 4 −- x = 0

⇒\Rightarrow x2 −- x(2h + 1) + 4 = 0 .... (ii)

For touching/tangency, Discriminant (D) = 0

i.e. (2h + 1)2 = 16 ⇒\Rightarrow 2h + 1 = ±\pm 4

⇒\Rightarrow 2h = ±\pm 4 −- 1

⇒\Rightarrow h=32,h=−52h = {3 \over 2},h = {{ - 5} \over 2} (Rejected) because part of circle lies outside R. So, h=32h = {3 \over 2} = radius of circle (c).

Putting h = 3/2 in Eq. (ii),

x2 −- 4x + 4 = 0 ⇒\Rightarrow (x −- 2)2 = 0 ⇒\Rightarrow x = 2

So, α\alpha = 2

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