Verified Solution • 2025–26 Session moderate

QJEEAdv21Ma685e: JEE Advanced 2021 Mathematics - Class 11 JEEAdv21Ma685e Solution: Consider the region R = {(x, y) ∈ R × R : x ≥ 0 and y 2 ≤ 4 −…

Consider the region R = {(x, y) ∈ R × R : x ≥ 0 and y2 ≤ 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, β) be a point where the circle C meets the curve y2 = 4 − x.

The radius of the circle C is ___________.

Step-by-Step Verified Solution

Given, x ≥ 0, y2 ≤ 4 − x

Let equation of circle be

(x − h)2 + y2 = h2 .... (i)

JEE Advanced 2021 Paper 2 Online Mathematics - Circle Question 7 English Explanation
Solving Eq. (i) with y2 = 4 − x, we get

x2 − 2hx + 4 − x = 0

⇒ x2 − x(2h + 1) + 4 = 0 .... (ii)

For touching/tangency, Discriminant (D) = 0

i.e. (2h + 1)2 = 16 ⇒ 2h + 1 = ± 4

⇒ 2h = ± 4 − 1

⇒ h=32,h=−52 (Rejected) because part of circle lies outside R. So, h=32 = radius of circle (C).

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