Let E denote the parabola y^2=8x. Let P=(-2,4 | vedclass

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

Question

(D) The parabola is $y^2=8x$,so $4a=8$,which gives $a=2$. The focus $F$ is $(2,0)$ and the directrix is $x=-2$.Point $P=(-2,4)$ lies on the directrix

Vedclass · Accepted Answer

$A, B, D$

Vedclass · Answer

(D) The parabola is $y^2=8x$,so $4a=8$,which gives $a=2$. The focus $F$ is $(2,0)$ and the directrix is $x=-2$.Point $P=(-2,4)$ lies on the directrix $x=-2$.It is a known property that the tangents drawn from a point on the directrix to a parabola are perpendicular to each other,and the chord of contact $QQ^{\prime}$ passes through the focus $F$.Since $PQ$ and $PQ^{\prime}$ are tangents from $P$ to the parabola,$\angle QPQ^{\prime} = 90^{\circ}$,so $(B)$ is $TRUE$.The chord of contact $QQ^{\prime}$ passes through the focus $F$,so $(D)$ is $TRUE$.The distance $PF = \sqrt{(2 - (-2))^2 + (0 - 4)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2}$. Thus,$(C)$ is $FALSE$.For a parabola,the angle subtended by the chord of contact at the focus is $180^{\circ}$ if the tangents are perpendicular,but the triangle $PFQ$ is right-angled at $F$ because the tangent at $Q$ makes an angle with the focal radius $FQ$ such that $\angle FQP = 90^{\circ}$ is not generally true,but here the geometry confirms $\angle PFQ = 90^{\circ}$ is not the case,however,$\angle FQP = 90^{\circ}$ is a property of tangents. Specifically,the triangle $PFQ$ is right-angled at $Q$ because the tangent at $Q$ is perpendicular to the line joining the focus to the point of contact. Thus,$(A)$ is $TRUE$.Therefore,the correct statements are $(A), (B),$ and $(D)$.

Similar Questions

For the parabola $y^2 = x$,let $PQ$ be a chord such that one endpoint $P$ is $(4, -2)$ and the chord is perpendicular to the axis of the parabola. What is the slope of the normal at $Q$?

The line $x - y + 2 = 0$ touches the parabola $y^2 = 8x$ at the point

Find the vertex,focus,axis,length of latus rectum,and the equation of the directrix for the parabola $y^{2} = 4x + 4y$.

An arch is in the form of a parabola with its axis vertical. The arch is $10 \, m$ high and $5 \, m$ wide at the base. How wide is it $2 \, m$ from the vertex of the parabola (in $, m$)?

Let $(x, y)$ be any point on the parabola $y^2 = 4x$. Let $P$ be a point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1 : 3$. Find the locus of $P$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module