At the point of intersection of the rectangul | vedclass

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

Question

(A) Let $(x_1, y_1)$ be the point of intersection. Then $y_1^2 = 4ax_1$ and $x_1y_1 = c^2$. For the parabola $y^2 = 4ax$,differentiating with respect

Vedclass · Accepted Answer

$	heta = 	an^{-1}(-2 	an \phi)$

Vedclass · Answer

(A) Let $(x_1, y_1)$ be the point of intersection. Then $y_1^2 = 4ax_1$ and $x_1y_1 = c^2$. For the parabola $y^2 = 4ax$,differentiating with respect to $x$ gives $2y \frac{dy}{dx} = 4a$,so $\frac{dy}{dx} = \frac{2a}{y}$. Thus,$	an \phi = \frac{2a}{y_1}$. For the hyperbola $xy = c^2$,differentiating with respect to $x$ gives $y + x \frac{dy}{dx} = 0$,so $\frac{dy}{dx} = -\frac{y}{x}$. Thus,$	an 	heta = -\frac{y_1}{x_1}$. Now,$\frac{	an 	heta}{	an \phi} = \frac{-y_1/x_1}{2a/y_1} = -\frac{y_1^2}{2ax_1}$. Substituting $y_1^2 = 4ax_1$,we get $\frac{	an 	heta}{	an \phi} = -\frac{4ax_1}{2ax_1} = -2$. Therefore,$	an 	heta = -2 	an \phi$,which implies $	heta = 	an^{-1}(-2 	an \phi)$.

At the point of intersection of the rectangular hyperbola $xy = c^2$ and the parabola $y^2 = 4ax$,tangents to the rectangular hyperbola and the parabola make an angle $\theta$ and $\phi$ respectively with the $X$-axis. Then:

Similar Questions

The triangle $PQR$ of area $A$ is inscribed in the parabola $y^2 = 4ax$ such that the vertex $P$ lies at the vertex of the parabola and the base $QR$ is a focal chord. The modulus of the difference of the ordinates of the points $Q$ and $R$ is:

If the tangent drawn to the parabola $y^2=4x$ at $(t^2, 2t)$ is the normal to the ellipse $4x^2+5y^2=20$ at $(\sqrt{5} \cos \theta, 2 \sin \theta)$,then

The quadratic equation whose roots are $l$ and $m$,where $l = \lim_{\theta \rightarrow 0} \left( \frac{3 \sin \theta - 4 \sin^2 \theta}{\theta} \right)$ and $m = \lim_{\theta \rightarrow 0} \frac{2 \tan \theta}{\theta(1 - \tan^2 \theta)}$,is:

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

If the normal to the rectangular hyperbola $xy = c^2$ at the point $t$ meets the curve again at $t_1$,then $t^3 t_1$ has the value equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module