InterviewSolution
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InterviewSolution
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The locus of the poles of the chords of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` which subtend a right angle at its centre isA. `(x^(2))/(a^(4))+(y^(2))/(b^(4))=1`B. `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/(a^(2))+(1)/(b^(2))`C. `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))`D. `(x^(2))/(a^(4))-(y^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))` |
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Answer» Let `P(h,k)` be the pole of a chord of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`. Then, the equation of the polar is `(hx)/(a^(2))-(ky)/(b^(2))=1`………`(i)` The combined equation of the lines joining the origin to the points of intersection of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` and the line `(i)` is `(x^(2))/(a^(2))-(y^(2))/(b^(2))=((hx)/(a^(2))-(ky)/(b^(2)))^(2)` Since the lines given by the above equation are at right angle. `:. "Coefficient of " x^(2)+ "Cofficient of" y^(2)=0` `implies(1)/(a^(2))-(h^(2))/(a^(4))-(1)/(b^(2))-(k^(2))/(b^(4))=0implies(h^(2))/(a^(4))+(k^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))` Hence, the locus of `P(h,k)` is `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/(a^(2))-(1)/(b^(2))` |
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