The angle between the parabolas y^2 = 4(x-1) | vedclass

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

Question

(A) For the parabola $y^2 = 4(x-1)$,the vertex is $(1, 0)$ and $a = 1$. The latus rectum is $x = 1+1 = 2$. The ends of the latus rectum are $(2, 2)$ a

Vedclass · Accepted Answer

$\frac{\pi}{2}$

Vedclass · Answer

(A) For the parabola $y^2 = 4(x-1)$,the vertex is $(1, 0)$ and $a = 1$. The latus rectum is $x = 1+1 = 2$. The ends of the latus rectum are $(2, 2)$ and $(2, -2)$.For the parabola $x^2 = -4(y-3)$,the vertex is $(0, 3)$ and $a = 1$. The latus rectum is $y = 3-1 = 2$. The ends of the latus rectum are $(2, 2)$ and $(-2, 2)$.The common point is $(2, 2)$.For $y^2 = 4(x-1)$,differentiating with respect to $x$: $2y \frac{dy}{dx} = 4 \implies \frac{dy}{dx} = \frac{2}{y}$. At $(2, 2)$,$m_1 = \frac{2}{2} = 1$.For $x^2 = -4(y-3)$,differentiating with respect to $x$: $2x = -4 \frac{dy}{dx} \implies \frac{dy}{dx} = -\frac{x}{2}$. At $(2, 2)$,$m_2 = -\frac{2}{2} = -1$.Since $m_1 	imes m_2 = 1 	imes (-1) = -1$,the tangents are perpendicular.Therefore,the angle between the parabolas is $\frac{\pi}{2}$.

The angle between the parabolas $y^2 = 4(x-1)$ and $x^2 + 4(y-3) = 0$ at the common end of their latus rectum is:

Similar Questions

If the straight line $y=mx+c$ is parallel to the axis of the parabola $y^2=lx$ and intersects the parabola at $\left(\frac{c^2}{8}, c\right)$,then the length of the latus rectum is

$A$ circle of radius $2$ unit passes through the vertex and the focus of the parabola $y^{2}=2x$ and touches the parabola $y=\left(x-\frac{1}{4}\right)^{2}+\alpha$,where $\alpha>0$. Then $(4\alpha-8)^{2}$ is equal to

If $x + y = k$ is a normal to the parabola ${y^2} = 12x$,then $k$ is

If two normals to the parabola $y^2 = 4x$ passing through the point $(15, 12)$ are $4x + y = 72$ and $3x - y = 33$,find the third normal.

Find the equation of the parabola whose axis is parallel to the $y$-axis and which passes through the points $(0,4), (1,9)$ and $(4,5)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module