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If the normal at any point `P` on the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` meets the axes at `Ga n dg,` respectively, then find the raio `P G: Pgdot`

Text Solution

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Let `P( a cos theta, b sin theta)` be a point on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
Then the quation of the normal at P is `ax sec theta-"by cosec" theta=a^(2)-b^(2)`
It meets the axe at
`G((a^(2)-b^(2))/(a)cos, theta,0)and g (0,-(a^(2)-b^(2))/(b) sin theta)`
`:. PG^(2)=(acos theta-(a^(2)-b^(2))/(a)costheta)^(2)+b^(2)sin^(2)theta=(b^(2))/(a^(2))(b^(2) cos^(2)theta +a^(2)sin^(2)theta)`
and `Pg^(2)=(a^(2))/(b^(2))(b^(2)cos^(2)theta +a^(2)sin^(2)theta)`
`:. PG ,Pg=b^(2):a^(2)`
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