InterviewSolution
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InterviewSolution
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A variable chord of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1,(b > a),`subtends a right angle at the center of the hyperbola if this chordtouches.a fixed circle concentric with the hyperbolaa fixed ellipse concentric with the hyperbolaa fixed hyperbola concentric with the hyperbolaa fixed parabola having vertex at (0, 0).A. a fixed circle concentric with the hyperbolaB. a fixed ellipse concentric with the hyperbolaC. a fixed hyperbola concentric with the hyperbolaD. a fixed parabola having vertex at (0, 0) |
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Answer» Correct Answer - A Let the variable chord be `x cos alpha+y sin alpha=p" (1)"` Let this chord intersect the hyperbola at A and B. Then the combined equation of OA and OB is given by `(x^(2))/(a^(2))-(y^(2))/(b^(2))=((x cos alpha+y sin alpha)/(p))^(2)` `x^(2)((1)/(a^(2))-(cos^(2)alpha)/(p^(2)))-y^(2)((1)/(b^(2))+(sin^(2)alpha)/(p^(2)))-(2 sin alpha cos alpha)/(p)xy=0` This chord subtends a right angle at the center. Therefore, `"Coefficient of " x^(2)+"Coefficient of "y^(2)=0` `"or "(1)/(a^(2))-(cos^(2)alpha)/(p^(2))-(1)/(b^(2))-(sin^(2) alpha)/(po)=0` `"or "(1)/(a^(2))-(1)/(b^(2))=(1)/(p^(2))` `"or "p^(2)=(a^(2)b^(2))/(b^(2)-a^(2))` Hence, p is constant, i.e., it touches the fixed circle. |
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