InterviewSolution
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InterviewSolution
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Consider the hyperbola `(X^(2))/(9)-(y^(2))/(a^(2))=1` and the circle `x^(2)+(y-3)=9`. Also, the given hyperbola and the ellipse `(x^(2))/(41)+(y^(2))/(16)=1` are orthogonal to each other. Combined equation of pair of common tangents between the hyperbola and the circle is given beA. `x^(2)-y^(2)=0`B. `x^(2)-9=0`C. `9y^(2)-19x^(2)=0`D. No common tangent. |
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Answer» Correct Answer - B Ellipse and hyperbola are orthogonal and therefore, they are confocal. `"So, "a_(h)e_(h)=a_(e)e_(e)` `rArr" "a^(2)+9=41-16` `rArr" "9+a^(2)=25` `rArr" "a^(2)=16` Thus, hyperbola is `(x^(2))/(9)-(y^(2))/(16)=1.` So, common tangents to the circle and hyperbola are `x = pm 3`. Director circle of hyperbola does not exist as `a lt b.` Director circle of circle is `x^(2)+(y-3)^(2)=18` `rArr" "x^(2)+y^(2)-6y-9=0` This meets the hyperbola `16x^(2)-9y^(2)=144` at four points from where tangents drawn to the circle `x^(2)+(y-3)^(2)=9` are perpendicular to each other. Let midpoint of AB be (h,k). So, equation of line AB is `hx+ky-3(y+k)=h^(2)+k^(2)-6k`. Since tangents at C and D intersect at the directrix, CD is the focal chord of hyperbola. So, AB passes through focus of the hyperbola and that is `(pm5,0)`. Therefore, required lacus is `x^(2)+y^(2)pm5x-3y=0`. |
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