InterviewSolution
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InterviewSolution
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The locus of a point, from where the tangents to the rectangularhyperbola `x^2-y^2=a^2`contain an angle of `45^0`, is`(x^2+y^2)^2+a^2(x^2-y^2)=4a^2``2(x^2+y^2)^2+4a^2(x^2-y^(2))=4a^2``(x^2+y^2)^2+4a^2(x^2-y^2)=4a^2``(x^2+y^2)+a^2(x^2-y^(2))=a^4`A. `(x^(2)+y^(2))^(2)+a^(2)(x^(2)-y^(2))=4a^(2)`B. `2(x^(2)+y^(2))^(2)+4a^(2)(x^(2)-y^(2))=4a^(2)`C. `(x^(2)+y^(2))^(2)+4a^(2)(x^(2)-y^(2))=4a^(4)`D. `(x^(2)+y^(2))^(2)+a^(2)(x^(2)-y^(2))=a^(4)` |
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Answer» Correct Answer - C Let `y=mx pm sqrt(m^(2)a^(2)-a^(2))` be two tangents that pass through (h,k). Then `(k-mh)^(2)=m^(2)a^(2)-a^(2)` `"or "m^(2)(h^(2)-a^(2))-2khm+k^(2)+a^(2)=0` `"or "m_(1)+m_(2)=(2kh)/(h^(2)-a^(2))` and `m_(1)m_(2)=(k^(2)+a^(2))/(h^(2)-a^(2))` Now, `tan45^(@)=(m_(1)-m_(2))/(1+m_(1)m_(2))` `"or "1=((m_(1)+m_(2))^(2)-4m_(1)m_(2))/((1+m_(1)m_(2))^(2))` `"or "(1+(k^(2)+a^(2))/(h^(2)-a^(2)))^(2)=((2kh)/(h^(2)-a^(2)))^(2)-4((k^(2)+a^(2))/(h^(2)-a^(2)))` `"or "(h^(2)+k^(2))^(2)=4h^(2)k^(2)-4(k^(2)+a^(2))(h^(2)-a^(2))` `"or "(x^(2)+y^(2))^(2)=4(a^(2)y^(2)-a^(2)x^(2)+a^(4))` `"or "(x^(2)+y^(2))^(2)+4a^(2)(x^(2)-y^(2))=4a^(4)` |
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