JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Circle

Consider the region

R = {(x, y) \in R \times R : x \geq 0

and

y^{2} \leq 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

y^{2} = 4 - x

.

The value of

α

is ____.

Answer: 2

Solution

From the previous question, the circle is

x^{2} + y^{2} - 3 x = 0

and it is tangent to the parabola

y^{2} = 4 - x

.

For the point of intersection, substitute

y^{2} = 4 - x

into the circle equation:

x^{2} + (4 - x) - 3 x = 0

x^{2} - 4 x + 4 = 0

(x - 2)^{2} = 0

x = 2

Therefore,

α = 2

JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Numerical answer

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