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

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(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

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$A, B, D$

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(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 $(D)$.

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