The radius of the smallest circle which touches the parabolas y=x²+2 and x=y²+2 is [3 Apr 2025 Shift 1]

Correct Answer is : 728\;{7{2}}{8}872

Correct Answer is : 728\;{7{2}}{8}872

Test Index

Parabola Part 2

Section: Mathematics

Question : 19 of 51

Marks: +1, -0

y=x^2+2

x=y^2+2

Solution:

The circle will have centre on

x=y

line since parabolas are symmetric about

y=x

line

The slope of tangent at closest point

y^2=x-2

\Rightarrow \;\; 2y \left(\;\frac{dy}{dx}\right) =1 \Rightarrow y=\;\frac{1}{2} \Rightarrow \;\text{ point will be }\; \left(\;\frac{9}{4}, \;\frac{1}{2}\right)

Similarly, on

x^2=y-2 \Rightarrow 2x = \;\frac{dy}{dx} = 1 \Rightarrow \left(\;\frac{1}{2}, \;\frac{9}{4}\right)

\;2r = \sqrt{ \left(\;\frac{9}{4} - \;\frac{1}{2}\right)^2 + \left(\;\frac{1}{2} - \;\frac{9}{4}\right)^2 } = \sqrt{2} \cdot \left| \;\frac{9}{4} - \;\frac{1}{2} \right|

\;=\;\frac{14}{8} \sqrt{2} = \;\frac{7\sqrt{2}}{4}

\;\Rightarrow r = \;\frac{7\sqrt{2}}{8}

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