The points (at_1^2, 2at_1),(at_2^2, 2at_2),an | vedclass

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(B) Three points $(x_1, y_1)$,$(x_2, y_2)$,and $(x_3, y_3)$ are collinear if the slope of the line segment connecting the first two points is equal to

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$t_1t_2 = -1$

Vedclass · Answer

(B) Three points $(x_1, y_1)$,$(x_2, y_2)$,and $(x_3, y_3)$ are collinear if the slope of the line segment connecting the first two points is equal to the slope of the line segment connecting the second and third points.Slope $m_1$ between $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$ is $\frac{2at_2 - 2at_1}{at_2^2 - at_1^2} = \frac{2a(t_2 - t_1)}{a(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_1 + t_2}$.Slope $m_2$ between $(at_2^2, 2at_2)$ and $(a, 0)$ is $\frac{0 - 2at_2}{a - at_2^2} = \frac{-2at_2}{a(1 - t_2^2)} = \frac{-2t_2}{1 - t_2^2}$.Equating the slopes: $\frac{2}{t_1 + t_2} = \frac{-2t_2}{1 - t_2^2}$.$1 - t_2^2 = -t_2(t_1 + t_2)$.$1 - t_2^2 = -t_1t_2 - t_2^2$.$1 = -t_1t_2$,which implies $t_1t_2 = -1$.

The points $(at_1^2, 2at_1)$,$(at_2^2, 2at_2)$,and $(a, 0)$ will be collinear,if

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