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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 ?


6 years ago

Jitender Singh
IIT Delhi
158 Points
							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 & RegardsJitender SinghIIT DelhiaskIITians Faculty

6 years ago
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### Course Features

• 731 Video Lectures
• Revision Notes
• Test paper with Video Solution
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Previous Year Exam Questions