The normal at a point P on the ellipse x2 + 4y2 = 16 meets the x-axis at Q.If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectum of the given ellipse at the points
A) \left(\pm \frac{3\sqrt{5}}{2}, \pm \frac{2}{7} \right) B) \left(\pm \frac{3\sqrt{5}}{2}, \pm \frac{\sqrt{19}}{4} \right) C)\left(\pm 2\sqrt{3}, \pm \frac{1}{7} \right) D)\left(\pm 2\sqrt{3}, \pm \frac{4\sqrt{3}}{7} \right)
1P≡(4cos@,2sin@)
Eqn of normal through P to the ellipse:4xsec@-2ycosec@=12.
the normal intersects the x-axis at (0,3/sec@)≡(0,3cos@)≡Q
Midpoint of PQ=(3.5cos@,sin@)
Let (x,y) be a point of locus, 2x/7=cos@
y=sin@
sin2@+cos2@=1.
so 4x2/49+y2=1........1
Latus rectum x=±2√3................2
solve 1 & 2
ans C.
