JEE Mathematics Online Test Ellipse and Hyperbola Set 01

[ { "question": "The equation \\( 2x^2 – 3xy + 5y^2 + 6x – 3y + 5 = 0 \\) represents", "options": [ "a pair of straight lines", "an ellipse", "a hyperbola", "a parabola" ], "answer": "b" }, { "question": "The set of real values of k for which the equation \\( (k + 1)x^2 + 2(k – 1)xy + y^2 – x + 2y + 3 = 0 \\) represents an ellipse is", "options": [ "\\( (3, +\\infty) \\)", "(0, 3)", "\\( (-\\infty, \\infty) \\)", "\\( (-\\infty, 0) \\)" ], "answer": "b" }, { "question": "The centre of the conic section \\( 14x^2 – 4xy + 11y^2 – 44x – 58y + 71 = 0 \\) is", "options": [ "(2, -3)", "(2, 3)", "(-2, 3)", "(-2, -3)" ], "answer": "b" }, { "question": "The eccentricity of the ellipse \\( \\frac{x^2}{4} + \\frac{y^2}{9} = 1 \\) is", "options": [ "\\( \\frac{\\sqrt{5}}{2} \\)", "\\( \\frac{\\sqrt{5}}{3} \\)", "\\( \\frac{2}{3} \\)", "\\( \\frac{4}{9} \\)" ], "answer": "b" }, { "question": "The eccentricity of the hyperbola \\( x^2 – 4y^2 = 16 \\) is", "options": [ "\\( \\frac{\\sqrt{3}}{2} \\)", "4", "\\( \\frac{\\sqrt{5}}{2} \\)", "2" ], "answer": "c" }, { "question": "The eccentricity of the conic section \\( 4(x^2 – y^2) = 1 \\) is", "options": [ "\\( \\sqrt{2} \\)", "2", "\\( \\frac{1}{4} \\)", "4" ], "answer": "a" }, { "question": "The latus rectum of the conic section \\( \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\) whose eccentricity = 3, is", "options": [ "\\( 2b(1 - e^2) \\)", "\\( \\frac{2b}{a^2} \\)", "\\( 2a(1 - e^2) \\)", "\\( \\frac{2a^2}{b} \\)" ], "answer": "c" }, { "question": "The ellipse \\( \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\) passes through the point (-3, 1) and has the eccentricity \\( \\sqrt{\\frac{2}{5}} \\). Then the major axis of the ellipse has the length", "options": [ "\\( 8\\sqrt{\\frac{2}{5}} \\)", "\\( 4\\sqrt{\\frac{2}{3}} \\)", "\\( 4\\sqrt{\\frac{2}{5}} \\)", "\\( 8\\sqrt{\\frac{2}{3}} \\)" ], "answer": "d" }, { "question": "The hyperbola \\( \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\) passes through the point (2, 3) and has the eccentricity 2. Then the transverse axis of the hyperbola has the length", "options": [ "3", "2", "1", "4" ], "answer": "b" }, { "question": "In the ellipse \\( x^2 + 3y^2 = 9 \\) the distance between the foci is", "options": [ "3", "\\( 2\\sqrt{6} \\)", "\\( \\sqrt{6} \\)", "\\( \\frac{2}{3}\\sqrt{6} \\)" ], "answer": "b" }, { "question": "The minor axis of the ellipse \\( 9x^2 + 5y^2 = 30y \\) is", "options": [ "\\( \\sqrt{5} \\)", "\\( \\sqrt{6} \\)", "\\( 2\\sqrt{5} \\)", "6" ], "answer": "c" }, { "question": "The foci of the ellipse \\( 25x^2 + 36y^2 = 225 \\) are", "options": [ "\\( \\left( 0, \\pm \\frac{1}{2}\\sqrt{11} \\right) \\)", "\\( \\left( \\pm \\frac{1}{2}\\sqrt{11}, 0 \\right) \\)", "\\( \\left( 0, \\pm \\frac{5}{2} \\right) \\)", "\\( \\left( \\pm \\frac{5}{2}, 0 \\right) \\)" ], "answer": "b" }, { "question": "If the eccentricity of the hyperbola \\( \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\) is e then the eccentricity of the hyperbola \\( \\frac{y^2}{b^2} - \\frac{x^2}{a^2} = 1 \\) is", "options": [ "e", "\\( e\\sqrt{e^2 - 1} \\)", "\\( e^2 – e \\)", "\\( \\frac{e}{\\sqrt{e^2 - 1}} \\)" ], "answer": "d" }, { "question": "If in an ellipse the minor axis = the distance between the foci and its latus rectum = 10 then the equation of the ellipse in the standard form is", "options": [ "\\( \\frac{x^2}{(10)^2} + \\frac{y^2}{(5\\sqrt{2})^2} = 1 \\)", "\\( \\frac{x^2}{(5\\sqrt{2})^2} + \\frac{y^2}{(10)^2} = 1 \\)", "\\( \\frac{x^2}{25} + \\frac{y^2}{(5/\\sqrt{2})^2} = 1 \\)", "none of the options" ], "answer": "a" }, { "question": "If in a hyperbola the eccentricity is \\( \\sqrt{3} \\), and the distance between the foci is 9 then the equation of the hyperbola in the standard form is", "options": [ "\\( \\frac{x^2}{\\left(\\frac{3\\sqrt{3}}{2}\\right)^2} - \\frac{y^2}{\\left(\\frac{3\\sqrt{2}}{2}\\right)^2} = 1 \\)", "\\( \\frac{x^2}{\\left(\\frac{3\\sqrt{3}}{2}\\right)^2} - \\frac{y^2}{\\left(\\frac{3\\sqrt{3}}{\\sqrt{2}}\\right)^2} = 1 \\)", "\\( \\frac{x^2}{\\left(\\frac{\\sqrt{3}}{2}\\right)^2} - \\frac{y^2}{\\left(\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)^2} = 1 \\)", "none of the options" ], "answer": "b" }, { "question": "If in an ellipse, a focus is (6, 7), the corresponding directrix is x + y + 2 = 0 and the eccentricity = \\( \\frac{1}{2} \\) then the equation of the ellipse is", "options": [ "\\( 9x^2 – 2xy + 9y^2 – 44x – 108y + 684 = 0 \\)", "\\( 7x^2 + 2xy + 7y^2 – 44x – 108y + 684 = 0 \\)", "\\( 7x^2 – 2xy + 7y^2 – 52x – 116y + 676 = 0 \\)", "none of the options" ], "answer": "c" }, { "question": "If for a rectangular hyperbola a focus is (1, 2) and the corresponding directrix is x + y = 1 then the equation of the rectangular hyperbola is", "options": [ "\\( x^2 – y^2 = 2 \\)", "\\( xy + y – 2 = 0 \\)", "\\( xy – y + 2 = 0 \\)", "none of the options" ], "answer": "b" }, { "question": "If two foci of an ellipse be (-2, 0) and (2, 0) and its eccentricity is \\( \\frac{2}{3} \\) then the ellipse has the equation", "options": [ "\\( 9x^2 + 5y^2 = 90 \\)", "\\( 5x^2 + 9y^2 = 45 \\)", "\\( 9x^2 + 5y^2 = 45 \\)", "\\( 5x^2 + 9y^2 = 90 \\)" ], "answer": "b" }, { "question": "If for a conic section a focus is (-1, 1), eccentricity = 3 and the equation of the corresponding directrix is x – y + 3 = 0 then the equation of the conic section is", "options": [ "\\( 7x^2 – 18xy + 7y^2 + 50x – 50y + 77 = 0 \\)", "\\( 7x^2 + 18xy + 7y^2 – 50x + 50y + 77 = 0 \\)", "\\( 7x^2 + 18xy + 7y^2 = 1 \\)", "none of the options" ], "answer": "a" }, { "question": "An ellipse having foci at (3, 1) and (1, 1) passes through the point (1, 3). Its eccentricity is", "options": [ "\\( \\sqrt{3} - 1 \\)", "\\( \\frac{1}{2}(\\sqrt{3} - 1) \\)", "\\( \\frac{1}{2}(\\sqrt{2} - 1) \\)", "\\( \\sqrt{2} - 1 \\)" ], "answer": "d" } ] const quizForm = document.getElementById("quiz-form"); mcqs.forEach((mcq, index) => { const questionDiv = document.createElement("div"); questionDiv.classList.add("question"); questionDiv.innerHTML = `

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