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question_answer1) A hyperbola has its centre at the origin, passes through the point \[(4,2)\] and has transverse axis of length 4 along the x-axis. If the eccentricity of the hyperbola is \[\frac{2}{\sqrt{a}},\] then the value of a is
question_answer2) If the tangent at point P on the circle \[{{x}^{2}}+{{y}^{2}}+6x+6y-2=0\] meets the straight line \[5x-2y+6=0\] at a point Q on the y-axis, then length PQ is
question_answer3) Let A \[(4,-4)\] and B \[(9,\,6)\] be points on the parabola, \[{{y}^{2}}=4x\]. Let C be chosen on the arc AOB of the parabola, where 0 is the origin, such that the area of \[\Delta ACB\] is maximum. Then, the area (in sq. units) of \[\Delta ACB\], is
question_answer4) If the circles \[{{x}^{2}}+{{y}^{2}}-2x-2y-7=0\] and \[{{x}^{2}}+{{y}^{2}}+4x+2y+k=0\] cut orthogonally. If the length of the common chord of the circles is \[\frac{m}{\sqrt{n}},\] then \[m+n\] is
question_answer5) The length of the chord of the parabola \[{{x}^{2}}=4y\]having equation \[x-\sqrt{2}y+4\sqrt{2}=0\] is \[a\sqrt{3},\] then the value of a is
question_answer6) An infinite number of tangents can be drawn from \[(1,2)\] to the cade \[{{x}^{2}}+{{y}^{2}}-2x-4y+\lambda =0\] then \[\lambda =\]
question_answer7) The equation of latus rectum of parabola is \[x+y=8\] and the equation of the tangent of the vertex is \[x+y=12\] and length of the latus rectum is \[a\sqrt{2},\] then the value of a is
question_answer8) If the area of the triangle whose one vertex is at the vertex of the parabola, \[{{y}^{2}}+4(x-{{a}^{2}})=0\] and the other two vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is
question_answer9) The radius of the circle passing through the foci of the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}=1\] and having its centre at \[(0,3)\] is
question_answer10) The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, \[y=12-{{x}^{2}}\]such that the rectangle lies inside the parabola, is
question_answer11) The equation of a tangent to the parabola \[{{y}^{2}}=8x\] is\[y=x+2\]. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is \[(-k,0)\] then the value of k is
question_answer12) The foci of the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and the hyperbola \[\frac{{{x}^{2}}}{144}-\frac{{{y}^{2}}}{81}=\frac{1}{25}\] coincide. Then the value of \[{{b}^{2}}\] is
question_answer13) In an ellipse, the distance between its foci is 6 and minor axis is 8. If its eccentricity is \[\frac{p}{q},\] then \[p+q\] is
question_answer14) An ellipse has OB as semi minor axis, F and F ' its focii and the angle FBF' is a right angle. If the eccentricity of the ellipse is \[\frac{1}{\sqrt{k}},\] then the value of k is
question_answer15) If PQ is focal chord of ellipse \[\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1\]which passes through \[S\equiv (3,0)\] and \[PS=2\] then length of chord PQ is
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