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JEE Advanced2009MathematicsEllipseActual

The normal at a point P on the ellipse x^2+4 y^2=16 meets the x -axis at Q . If M is the mid-point of the line segment P Q , then the locus of M intersects the latusrectum of the given ellipse at the points

Options

  1. A( 3 5 2 , 2 7 )
  2. B( 3 5 2 , 19 4 )
  3. C( 2 3 , 1 7 )
  4. D( 2 3 , 4 3 7 )

Correct answer

C. ( 2 3 , 1 7 )

Step-by-step solution

Given, x^2 16 + y^2 4 =1 Here, a=4, b=2 Equation of normal is aligned & 4 x -2 y cosec =12 & M ( 7 2 , )=(h, k) (say) & h= 7 2 = 2 h 7 & [ ^2 + ^2 =1 ] & aligned Hence, locus is 4 x^2 49 +y^2=1 For given ellipse, e^2=1- 4 16 = 3 4 array rc e= 3 2 x= 4 3 2 = 2 3 & [ x= a e] (ii) array On solving Eqs. (i) and (ii), we get aligned & 4 49 12+y^2=1 y^2=1- 48 49 = 1 49 & y= 1 7 aligned Required points ( 2 3 , 1 7 ) .

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