The coordinates of the ends of a focal chord | vedclass

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(B) Let the parameters of the endpoints of the focal chord be $t_1$ and $t_2$. For a parabola $y^2 = 4ax$,the coordinates are $(at_1^2, 2at_1)$ and $(

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$-3a^2$

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(B) Let the parameters of the endpoints of the focal chord be $t_1$ and $t_2$. For a parabola $y^2 = 4ax$,the coordinates are $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$. Since it is a focal chord,the product of the parameters is $t_1t_2 = -1$. We have $x_1 = at_1^2$ and $x_2 = at_2^2$,so $x_1x_2 = a^2(t_1t_2)^2 = a^2(-1)^2 = a^2$. We have $y_1 = 2at_1$ and $y_2 = 2at_2$,so $y_1y_2 = 4a^2(t_1t_2) = 4a^2(-1) = -4a^2$. Therefore,$x_1x_2 + y_1y_2 = a^2 - 4a^2 = -3a^2$.

The coordinates of the ends of a focal chord of a parabola $y^2 = 4ax$ are $(x_1, y_1)$ and $(x_2, y_2)$. Then $x_1x_2 + y_1y_2$ has the value equal to

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