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AP EAMCET201923 Apr 2019Morning ShiftMathematicsEllipseActual

Let a tangent drawn at any point on the ellipse ( x^2 25 + y^2 16 =1 ) cut the (X )-axis at (Q ). Let (R ) be the image of (Q ) with respect to (y=x ). If (S ) is a circle with (Q R ) as its diameter, then the fixed point through which the circle (S ) passes is

Options

  1. A((5,4) )
  2. B((4,5) )
  3. C((0,0) )
  4. D((0,5) )

Correct answer

C. ((0,0) )

Step-by-step solution

The equation of given ellipse is ( x^2 25 + y^2 16 =1 ). Let a point (P(5 , 4 ) ) on the ellipse, then equation of tangent to the ellipse at point (P ) is (x ( 5 )+y ( 4 )=1 ) ...(i) So, point (Q ( 5 , 0 ) ) Point (R(0,5 ) ) Now, equation of circle with (Q R ) as its diameter is (x(x-5 )+y(y-5 )=0 ) ( x^2+y^2-(5 ) x-(5 ) y=0 ) The above circle passes through the origin. Hence, option (c) is correct.

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