InterviewSolution
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InterviewSolution
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Prove that the locus of the point of intersection of the tangents atthe ends of the normal chords of the hyperbola `x^2-y^2=a^2`is `a^2(y^2-x^2)=4x^2y^2dot` |
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Answer» Normal at a point `(a sec theta, a tan theta)` is `x cos theta+y cot theta=2a` If `P(x_(1),y_(1))` is the point of intersection of the tangents at the ends of normal chord (1), then (1) must be the chord of contact of P(h, k) whose equation is given by `hx-ky=a^(2)" (2)"` Comparing (1) and (2) and eliminating `theta`, we get `(a^(2))/(4h^(2))-(a^(2))/(4k^(2))=1` Hence, the locus is `(1)/(x^(2))-(1)/(y^(2))=(4)/(a^(2))` |
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