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question_answer1) Find the number of real tangents that can be drawn to the ellipse \[3{{x}^{2}}+\text{ }5{{y}^{2}}=\text{ }32\] passing through \[\left( 3,\text{ }5 \right).\]
question_answer2) A man running round a race course notes that the sum of the distances of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 meters. The area of the path he encloses is \[k\pi \] then find k.
question_answer3) An ellipse having foci at (3, 1) and (1, 1) passes through the point (1, 3). Then find its eccentricity.
question_answer4) If the eccentricity of an ellipse be 5/8 and the distance between its foci be 10, then find its latus rectum.
question_answer5) Suppose S and S' are foci of the ellipse \[\frac{{{x}^{2}}}{25}=\frac{{{y}^{2}}}{16}=1\]. If P is variable point on the ellipse and, if \[\Delta \] is area of the triangle PSS', then find the maximum value of \[\Delta \].
question_answer6) Tangent at a point P on \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] meets the x-axis at A and y-axis at B. The locus of the midpoint of AB is \[\frac{{{a}^{2}}}{{{x}^{2}}}+\frac{{{b}^{2}}}{{{y}^{2}}}=k,\] then find k.
question_answer7) Find the radius of the circle passing through the foci of ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}=1,\]and having its center (0, 3).
question_answer8) \[{{S}_{1}},\,\,{{S}_{2}}\] are foci of an ellipse of major axis of length 10 units and P is any point on the ellipse such that perimeter of \[\Delta P{{S}_{1}}\,{{S}_{2}}\] is 15. Find the eccentricity of ellipse.
question_answer9) If extremities of diameter of the circle \[{{x}^{2}}+{{y}^{2}}=16\] are foci of an ellipse, then find the eccentricity of the ellipse, if its size is just sufficient to contain the circle.
question_answer10) If B and B' are the ends of minor axis and S and S' are the foci of the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{9}=1,\] then find the area of rhombus SBS'B' formed.
question_answer11) If the eccentricity of the two ellipse \[\frac{{{x}^{2}}}{169}+\frac{{{y}^{2}}}{25}=1\] and \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are equal, then find the value of \[\frac{a}{b}\].
question_answer12) For a point P on the ellipse \[9{{x}^{2}}+36{{y}^{2}}=324,\] with foci S and S', find value of \[SP+S'P.\]
question_answer13) Find the number of values of c such that the straight line \[y=4x+c~~\]touches the curve \[{{x}^{2}}/4~+{{y}^{2}}=1.\]
question_answer14) If the eccentricity of the ellipse \[4{{x}^{2}}+9{{y}^{2}}-8x-6y+1=0\] is \[\frac{\sqrt{5}}{k}\]then find k.
question_answer15) Find the product of the of length perpendiculars drawn from the foci upon any tangent to the ellipse\[3{{x}^{2}}+4{{y}^{2}}=12\].
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