Question

{"given":"The equation of the conic is $\\frac{x^2}{36-\\lambda} + \\frac{y^2}{25-\\lambda} = 1$, where $\\lambda < 25$.","key_observation":"For the given ellipse, $a^2 = 36-\\lambda$ and $b^2 = 25-\\lambda$. We need to determine which of the given parameters do not depend on the value of $\\lambda$.","option_analysis":[{"label":"(A)","text":"Equation of directrices","verdict":"incorrect","explanation":"The eccentricity is $e = \\sqrt{1 - \\frac{b^2}{a^2}} = \\sqrt{1 - \\frac{25-\\lambda}{36-\\lambda}} = \\frac{\\sqrt{11}}{\\sqrt{36-\\lambda}}$. The directrices are $x = \\pm \\frac{a}{e} = \\pm \\frac{36-\\lambda}{\\sqrt{11}}$, which depends on $\\lambda$."},{"label":"(B)","text":"Difference between the squares of the lengths of major and minor axes","verdict":"correct","explanation":"The difference between the squares of the lengths of the major and minor axes is $(2a)^2 - (2b)^2 = 4(a^2 - b^2) = 4(36-\\lambda - (25-\\lambda)) = 44$, which is a constant and independent of $\\lambda$."},{"label":"(C)","text":"Coordinates of foci","verdict":"correct","explanation":"The coordinates of the foci are $(\\pm ae, 0)$. Since $a^2 e^2 = a^2 - b^2 = 11$, we have $ae = \\sqrt{11}$. Thus, the foci are $(\\pm \\sqrt{11}, 0)$, which are independent of $\\lambda$."},{"label":"(D)","text":"Eccentricity","verdict":"incorrect","explanation":"The eccentricity is $e = \\frac{\\sqrt{11}}{\\sqrt{36-\\lambda}}$, which clearly depends on the value of $\\lambda$."}],"answer":"(B), (C)","formula_steps":["","",""]}