Q.277. Largest circle inscribed in an ellipse.

Question 277.
Find the equation of the largest circle with centre (1,0) that can be inscribed in the ellipse x^2 + 4y^2 =16.

Answer 277.
Let the required circle have radius r
=> its eqn. is
(x - 1)^2 + y^2 = r^2.
Solving it with the eqution of the ellipse
x^2 + 4y^2 = 16
=> (x - 1)^2 + (16 - x^2) / 4 = r^2
=> 4(x^2 - 2x + 1) + 16 - x^2 = 4r^2
=> 3x^2 - 8x + 20 - 4r^2 = 0.

As the circle and ellipse touch each other, discriminant of the quadratic eqn. in x should be zero.
=> 8^2 - 4 * 3 * (20 - 4r^2) = 0
=> 64 - 48 (5 - r^2) = 0
=> 4 - 3 (5 - r^2) = 0
=> 3r^2 = 11
=> r^2 = 11/3

=> eqn. of the circle is
(x - 1)^2 + y^2 = 11/3
=> 3x^2 + 3y^2 - 6x - 8 = 0

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