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JEE Advanced 2021 Paper 2

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Consider the region

R=\{(x,y)\in\mathbb{R}\times\mathbb{R}: x\ge 0

and

y^{2}\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

(\alpha, \beta)

be a point where the circle

C

meets the curve

y^{2}=4-x

Section: Mathematics

Question : 45 of 57

Marks: +1, -0

C

Solution:

Since

C

be the circle that has largest radius so, it touches the

y

-axis at

(0,0)

and centre at

x

-axis.

Let the equation of circle be

x^{2}+y^{2}+\lambda x=0

For point of intersection of circle\;parabola

y^{2}=4-x

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

\Rightarrow x^{2}+x(\lambda-1)+4=0\;\;\ldots

(i)

For tangency:

\Delta=0

\Rightarrow(\lambda-1)^{2}-16=0

\Rightarrow \lambda=5

(rejected) or

\lambda=-3

\therefore\;\;

Equation of circle

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

Radius

=\;\frac{3}{2}=1.5

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