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Byju's Answer

Standard XII

Mathematics

Focii of Ellipse

A chord PP'...

Question

A chord $P P^{'}$ of a parabola cuts the axis of the parabola at $O$ . The feet of the perpendiculars from $P$ and $P^{'}$ on the axis are $M$ and $M^{'}$ respectively. If $V$ is the vertex then $V M$ , $V O$ , $V M^{'}$ are in

A P

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G P

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H P

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none of these

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Solution

The correct option is B $G P$
Let the parabola be $y^{2} = 4 a x$ and $P = (a t_{1}^{2}, 2 a t_{1}), P^{'} = (a t_{2}^{2}, 2 a t_{2})$
$∴ V M = a t_{1}^{2}, V M^{'} = a t_{2}^{2}$ and $V O = k$
where $∣ ∣ ∣ ∣ ∣ \begin{matrix} a t_{1}^{2} & 2 a t_{1} & 1 a t_{2}^{2} & 2 a t_{2} & 1 k & 0 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow k = a t_{1} t_{2} = 0 ∴ V M . V M^{'} = {(a t_{1} t_{2})}^{2} = k^{2} = {O V}^{2}$

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$\begin{matrix} Column 1 & Column 2 P) Focal distance & T) Focal chord perpendicular to axis Q) Double ordinate & U) A chord perpendicular to axis R) Latus rectum & V) Distance of a point on parabola from directrix S) Vertex & W) Meeting point of axis and parabola \end{matrix}$

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A chord PP′ of a parabola cuts the axis of the parabola at O. The feet of the perpendiculars from P and P′ on the axis are M and M′ respectively. If V is the vertex then VM,VO,VM′ are in

The correct option is B GPLet the parabola be y2=4ax and P=(at21,2at1),P′=(at22,2at2)∴VM=at21,VM′=at22 and VO=kwhere ∣∣ ∣ ∣∣at212at11at222at21k01∣∣ ∣ ∣∣=0⇒k=at1t2=0∴VM.VM′=(at1t2)2=k2=OV2

A chord $P P^{'}$ of a parabola cuts the axis of the parabola at $O$ . The feet of the perpendiculars from $P$ and $P^{'}$ on the axis are $M$ and $M^{'}$ respectively. If $V$ is the vertex then $V M$ , $V O$ , $V M^{'}$ are in