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11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer
    The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{5}=1\], is [IIT Screening 2003]

    A)            27/4 sq. unit                             

    B)            9 sq. unit

    C)            27/2 sq. unit                             

    D)            27 sq. unit

    Correct Answer: D

    Solution :

               By symmetry the quadrilateral is a rhombus. So area is four times the area of the right angled triangle formed by the tangent and axes in the Ist quadrant. Now, \[ae=\sqrt{{{a}^{2}}-{{b}^{2}}}\Rightarrow ae=2\]            Þ Tangent (in first quadrant) at end of latus rectum \[\left( 2,\frac{5}{3} \right)\] is \[\frac{2}{9}x+\frac{5}{3}\frac{y}{5}=1\] i.e., \[\frac{x}{9/2}+\frac{y}{3}=1\]                    Area \[=4.\,\frac{1}{2}.\,\frac{9}{2}.3=27\] sq. unit.


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