The minimum distance between the parabola y^2 | vedclass

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

Question

(C) The parabola is $y^2 = 8x$,so $4a = 8$,which gives $a = 2$. The general point on the parabola is $P(2t^2, 4t)$.The slope of the tangent at $P$ is

Vedclass · Accepted Answer

$4\sqrt{2}$

Vedclass · Answer

(C) The parabola is $y^2 = 8x$,so $4a = 8$,which gives $a = 2$. The general point on the parabola is $P(2t^2, 4t)$.The slope of the tangent at $P$ is given by $\frac{dy}{dx} = \frac{4}{2t} = \frac{2}{t}$.For the minimum distance,the tangent at $P$ must be parallel to the line $x + y + 4 = 0$,which has a slope of $-1$.Setting $\frac{2}{t} = -1$,we get $t = -2$.The point $P$ is $(2(-2)^2, 4(-2)) = (8, -8)$.The distance from point $P(8, -8)$ to the line $x + y + 4 = 0$ is $d = \frac{|8 - 8 + 4|}{\sqrt{1^2 + 1^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}$.The minimum distance between the parabola and its image is twice the distance from the point $P$ to the line of reflection,which is $2 	imes (2\sqrt{2}) = 4\sqrt{2}$.

The minimum distance between the parabola $y^2 = 8x$ and its image with respect to the line $x + y + 4 = 0$ is:

Similar Questions

The common tangent to the parabolas $y^2=32x$ and $x^2=256y$ is:

Consider the parabola with vertex $\left(\frac{1}{2}, \frac{3}{4}\right)$ and the directrix $y=\frac{1}{2}$. Let $P$ be the point where the parabola meets the line $x=-\frac{1}{2}$. If the normal to the parabola at $P$ intersects the parabola again at the point $Q$,then $(PQ)^{2}$ is equal to :

The tangents to the parabola $y^2 = 4ax$ make angles $\theta_1$ and $\theta_2$ with the positive $x$-axis. If $\cot \theta_1 + \cot \theta_2 = c$,then the locus of their point of intersection is

The coordinates of a point on the parabola $y^2 = 8x$ whose focal distance is $4$ are

$A$ particle is moving in the $xy$-plane along a curve $C$ passing through the point $(3, 3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $PQ$,then $C$ is a parabola with

Vedclass Test Series

Exam Paper Generator

Online Exam Module