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Normal to the ellipse (x^2)/(84)+(y^2)/(...

Normal to the ellipse `(x^2)/(84)+(y^2)/(49)=1` intersects the major and minor axes at `Pa n dQ` , respectively. Find the locus of the point dividing segment `P Q` in the ratio 2:1.

Text Solution

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The equation of normal is `8x sec thea-7y=15`
It meets the exes at
`P((15)/(8)costheta,0) and Q(0,(-15)/(7)sin theta)`
Let M(h,k) divides PQ in ratio `2:1` Thefefore,
`3h=(15)/(8)cos theta, 3k=-(30)/(7)sin theta`
`:. Cos theta=(8h)/(5),sin theta=(-7k)/(10)`

Eliminating `theta`m we get the locus as
`(64x^(2))/(25)+(49y^(2))/(100)=1`
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