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JEE Main 2020
Conic Sections
Parabola
Easy

Question

Let P\mathrm{P} and Q\mathrm{Q} be any points on the curves (x−1)2+(y+1)2=1(x-1)^{2}+(y+1)^{2}=1 and y=x2y=x^{2}, respectively. The distance between PP and QQ is minimum for some value of the abscissa of PP in the interval :

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Solution

Key Concept: Minimum Distance Between a Circle and a Curve

The problem asks for the minimum distance between a point P on a circle and a point Q on a parabola. A fundamental principle in geometry for finding the minimum distance between a convex curve (like a circle) and another curve (like a parabola) is that the line segment connecting the two points of minimum distance must be a common normal to both curves at those points.

  1. Normal to a Circle: For any point P on a circle, the normal line to the circle at P always passes through the center of the circle, C.
  2. Common Normal Condition: Therefore, the line segment PQ, representing the minimum distance, must be normal to the parabola at Q and also normal to the circle at P. This implies
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