InterviewSolution
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InterviewSolution
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A point `P`moves such that the chord of contact of the pair of tangents from `P`on the parabola `y^2=4a x`touches the rectangular hyperbola `x^2-y^2=c^2dot`Show that the locus of `P`is the ellipse `(x^2)/(c^2)+(y^2)/((2a)^2)=1.` |
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Answer» The equation of the chord of contact of parabola w.r.t. `P(x_(1),y_(1))` is given by `yy_(1)=2a(x+x_(1))" (1)"` Equation (1) touches the curve `x^(2)-y^(2)=c^(2)" (2)"` Using the condition of tangency on `y=(2ax)/(y_(1))+(2ax_(1))/(y_(1))` we get `(4a^(2)x_(1)^(2))/(y_(1))=c^(2)(4a^(2))/(y_(1)^(2))-c^(2)" "[because"(1) is tangent to"(x^(2))/(c^(2))-(y^(2))/(c^(2))=1]` `"or "4a^(2)x_(1)^(2)=4a^(2)c^(2)-c^(2)y_(1)^(2)` `"or "4a^(2)x_(1)^(2)+c^(2)y_(1)^(2)=4a^(2)c^(2)` Hence, the locus is `(x^(2))/(c^(2))+(y^(2))/((2a)^(2))=1` which is an ellipse. |
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