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Conic Sections
Ellipse
Medium

Question

Let C be the largest circle centred at (2, 0) and inscribed in the ellipse x236+y216=1{{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1. If (1, α\alpha) lies on C, then 10 α2\alpha^2 is equal to ____________

Answer: 2

Solution

1. Key Concept: Normal to the Ellipse and Tangency

For the largest circle centered at a fixed point (h,k)(h, k) to be inscribed within an ellipse, the circle must be tangent to the ellipse. At the point(s) of tangency (x1,y1)(x_1, y_1), a fundamental geometric property holds: The normal to the ellipse at (x1,y1)(x_1, y_1) must pass through the center of the circle (h,k)(h, k).

Why this is true:

  • At the point of tangency, the circle and the ellipse share a common tangent line.
  • The radius of the circle drawn to the point of tangency is perpendicular to this common tangent line. This radius is effectively the normal to the circle at that
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