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through the vertex of the parabola y^2=4x chords op and oq are drawn at right angles to one another.the locus of mid point of pq is ?

through the vertex of the parabola y^2=4x chords op and oq are drawn at right angles to one another.the locus of mid point of pq is ?

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1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans:
P & Q are points on parabola. Let the coordinates of P & Q be
P(t_{1}, 2t_{1}^{2}), Q(t_{2}, 2t_{2}^{2})
Slope(OP) = \frac{2}{t_{1}}
Slope(OQ) = \frac{2}{t_{2}}
Since OP & OQ are perpendicular, we have
\frac{2}{t_{1}}.\frac{2}{t_{2}} = -1
t_{1}.t_{2} = -4
Let mid-point of P & Q be h, k
h = \frac{t_{1}^{2} + t_{2}^{2}}{2}, k = t_{1} + t_{2}
2h = (t_{1}+t_{2})^{2} - 2t_{1}t_{2}
t_{1} + t_{2} = k, t_{1}._{2} = -4
2h = k^{2} + 8
h \rightarrow x, k\rightarrow y
Locus is
y^{2} = 2x - 8
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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