The total number of tangents through the poin | vedclass

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(C) For the ellipse $E_1: 3x^2 + 5y^2 = 32$,check the position of the point $(3,5)$ by substituting it into the equation: $3(3)^2 + 5(5)^2 = 3(9) + 5(

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$3$

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(C) For the ellipse $E_1: 3x^2 + 5y^2 = 32$,check the position of the point $(3,5)$ by substituting it into the equation: $3(3)^2 + 5(5)^2 = 3(9) + 5(25) = 27 + 125 = 152$. Since $152 > 32$,the point $(3,5)$ lies outside the ellipse $E_1$. Thus,$2$ tangents can be drawn to $E_1$.For the ellipse $E_2: 25x^2 + 9y^2 = 450$,check the position of the point $(3,5)$: $25(3)^2 + 9(5)^2 = 25(9) + 9(25) = 225 + 225 = 450$. Since the result equals $450$,the point $(3,5)$ lies on the ellipse $E_2$. Thus,only $1$ tangent can be drawn to $E_2$.The total number of tangents is $2 + 1 = 3$.

The total number of tangents through the point $(3,5)$ that can be drawn to the ellipses $3x^2 + 5y^2 = 32$ and $25x^2 + 9y^2 = 450$ is

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