Find the number of real tangents that can be | vedclass

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(C) point $(x_1, y_1)$ lies outside an ellipse if the power of the point with respect to the ellipse is positive.For the first ellipse $E_1: 3x^2 + 5y

Vedclass · Accepted Answer

$3$

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(C) point $(x_1, y_1)$ lies outside an ellipse if the power of the point with respect to the ellipse is positive.For the first ellipse $E_1: 3x^2 + 5y^2 - 32 = 0$,at $(3, 5)$:$S_1 = 3(3)^2 + 5(5)^2 - 32 = 3(9) + 5(25) - 32 = 27 + 125 - 32 = 120 > 0$.Since $S_1 > 0$,the point $(3, 5)$ lies outside $E_1$,so $2$ tangents can be drawn to $E_1$.For the second ellipse $E_2: 25x^2 + 9y^2 - 450 = 0$,at $(3, 5)$:$S_2 = 25(3)^2 + 9(5)^2 - 450 = 25(9) + 9(25) - 450 = 225 + 225 - 450 = 0$.Since $S_2 = 0$,the point $(3, 5)$ lies on the ellipse $E_2$,so only $1$ tangent can be drawn to $E_2$.The total number of real tangents that can be drawn from $(3, 5)$ to both ellipses is $2 + 1 = 3$.

Find the number of real tangents that can be drawn from the point $(3, 5)$ to the ellipses $3x^2 + 5y^2 = 32$ and $25x^2 + 9y^2 = 450$.

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