The ratio between the length of the subtangen | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

$2:1$

Vedclass · Answer

(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:

Similar Questions

What is the equation of the tangent to the curve $y = 2 \cos x$ at $x = \pi / 4$?

If the tangent to the curve $xy + ax + by = 0$ at $(1, 1)$ makes an angle $\tan^{-1}(2)$ with the $x$-axis,then $\left( \frac{a + b}{ab} \right)$ is equal to:

An angle between the curves $x^2-y^2=4$ and $x^2+y^2=4 \sqrt{2}$ is

If $y=2x$ is a tangent to the curve $y^2=ax^3+b$ at $(1,2)$,then $(a, b)=$

The tangent to a non-linear curve $y = f(x)$ at any point $P(x, y)$ intersects the $x$-axis and $y$-axis at $A$ and $B$ respectively. If the normal to the curve $y = f(x)$ at $P$ intersects the $y$-axis at $C$ such that $AC = BC$,and $f(2) = 3$,then the equation of the curve is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module