The ratio between the length of the subtangen | vedclass

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(D) Let $(x_1, y_1)$ be any point other than the origin on the parabola $y^2 = 16ax$. The length of the subtangent at $(x_1, y_1)$ is given by $y_1 \l

Vedclass · Accepted Answer

$2:1$

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(D) Let $(x_1, y_1)$ be any point other than the origin on the parabola $y^2 = 16ax$. The length of the subtangent at $(x_1, y_1)$ is given by $y_1 \left| \frac{dx}{dy} \right|_{(x_1, y_1)}$. Differentiating $y^2 = 16ax$ with respect to $x$, we get $2y \frac{dy}{dx} = 16a$, which implies $\frac{dy}{dx} = \frac{8a}{y}$. Therefore, $\frac{dx}{dy} = \frac{y}{8a}$. The length of the subtangent at $(x_1, y_1)$ is $y_1 \left( \frac{y_1}{8a} \right) = \frac{y_1^2}{8a}$. Since the point $(x_1, y_1)$ lies on the parabola, $y_1^2 = 16ax_1$. Substituting this into the expression for the length of the subtangent, we get $\frac{16ax_1}{8a} = 2x_1$. The ratio of the length of the subtangent to the abscissa $x_1$ is $\frac{2x_1}{x_1} = 2:1$.

The ratio between the length of the subtangent at any point other than the origin on the parabola $y^2 = 16ax$ and the abscissa of that point is:

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