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JEE Advanced 2021 · Paper 2, Mathematics question 7

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Question Stem for Question Nos. 7 and 8

Consider the region R={(x,y)∈R×R:x≥0 and y2≤4−x}R = \{(x, y) \in \mathbb R \times \mathbb R : x \ge 0 \text{ and } y^2 \le 4 - x\}. Let F\mathcal F be the family of all circles that are contained in RR and have centers on the xx-axis. Let CC be the circle that has largest radius among the circles in F\mathcal F. Let (α,β)(\alpha, \beta) be a point where the circle CC meets the curve y2=4−xy^2 = 4 - x.

The radius of the circle CC is ____.

Answer and worked solution

Official answer: 1.49

Distance from the centre to the parabola. For a circle centred at (h,0)(h,0), the squared distance to a point of y2=4−xy^2=4-x is (x−h)2+4−x(x-h)^2+4-x. It is smallest at x=h+12x=h+\frac12, where it equals 154−h\frac{15}4-h. So the largest radius that fits inside the parabola is r2=154−hr^2=\frac{15}4-h.

Staying in x≥0x\ge0. We also need r≤hr\le h. Since rr shrinks as hh grows, the best circle has r=hr=h:
h2=154−h  ⟹  h=32.h^2=\frac{15}4-h\implies h=\frac32.

So the radius of CC is 1.51.5.

Topics:Equation of a circleParabolaMaxima and minima
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