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question_answer1) Consider the set of hyperbola \[xy=k,\,\,\,k\in R.\] Let \[{{e}_{1}}\] be the eccentricity when \[k=4\] and \[{{e}_{2}}\] be the eccentricity when \[k=9\] then \[{{e}_{1}}-{{e}_{2}}=\]
question_answer2) If the foci of the ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and the hyperbola \[\frac{{{x}^{2}}}{144}-\frac{{{y}^{2}}}{81}=\frac{1}{25}\] coincide, then find the value \[{{b}^{2}}.\]
question_answer3) Find the eccentricity of the conjugate hyperbola of the hyperbola \[{{x}^{2}}-3{{y}^{2}}=1.\]
question_answer4) A tangent drawn to hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] at \[P\left( \frac{\pi }{6} \right)\] forms a triangle of area \[3{{a}^{2}}\] square units, with coordinate axes, then the square of its eccentricity.
question_answer5) Find the eccentricity of the hyperbola \[\frac{\sqrt{1999}}{3}\left( {{x}^{2}}-{{y}^{2}} \right)=1.\]
question_answer6) Find the eccentricity of the hyperbola whose latus rectum 12 and semi-conjugate axis is\[2\sqrt{3}\].
question_answer7) If the eccentricity and length of latus rectum of a hyperbola are \[\frac{\sqrt{13}}{3}\]and \[\frac{10}{3}\]units respectively, then find the length of the transverse axis.
question_answer8) If P is any point on the hyperbola \[\frac{{{\left( x-1 \right)}^{2}}}{9}-\frac{{{\left( y+1 \right)}^{2}}}{16}=1\] and \[{{S}_{1}}\] and \[{{S}_{2}}\] are its foci then find value of \[\left| {{S}_{1}}P-{{S}_{2}}P \right|.\]
question_answer9) The straight line \[x+y=\sqrt{2}p\] touches the hyperbola \[4{{x}^{2}}-\text{ }9{{y}^{2}}=36,\] then find value of \[{{p}^{2}}.\]
question_answer10) If e and e' are the eccentricities of the ellipse \[5{{x}^{2}}+9{{y}^{2}}=45\] and the hyperbola \[5{{x}^{2}}-4{{y}^{2}}=45\] respectively, then find value of \[ee'.\]
question_answer11) If the latus rectum of a hyperbola subtends \[60{}^\circ \]angle at the other focus, then find its eccentricity.
question_answer12) If distance between directrices of a rectangular hyperbola is 10, then find distance between its foci.
question_answer13) Find the length of the latus rectum of the hyperbola \[xy-3x-3y+7=0.\]
question_answer14) Let LL' be the latus rectum through the focus of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and A' be the farther vertex If \[\Delta \,\,A'LL'\] is equilateral and the eccentricity of the hyperbola (axes are coordinate axes) is \[\left( 1+\frac{1}{k} \right)\] then find k.
question_answer15) Find the eccentricity of the hyperbola \[9{{x}^{2}}-16{{y}^{2}}+72x-32y-16=0.\]
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