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JEE Main202529 Jan 2025Evening ShiftMathematicsParabolaActual

Let y^2=12 x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that ( SP )( SQ )= 147 4 . Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64 x^2+64 y^2- x-64 3 y= , then - is equal to ________.

Correct answer

0

Step-by-step solution

y ^2=12 x a =3 SP SQ = 147 4 Let P (3 t ^2, 6 t ) and t ₁ t ₂=-1 (ends of focal chord) So, Q ( 3 t^2 , -6 t ) aligned & S (3,0) & SP SQ = PM ₁ QM ₂ aligned (dist. from directrix) aligned & = (3+3 t ^2 ) (3+ 3 t ^2 )= 147 4 & (1+ t ^2 )^2 t ^2 = 49 12 & t ^2= 3 4 , 4 3 & t = 3 2 , 2 3 aligned considering t = - 3 2 P ( 9 4 ,-3 3 ) and Q (4,4 3 ) Hence, diametric circle: aligned & (x-4) (x- 9 4 )+(y+3 3 )(y-4 3 )=0 & x^2+y^2- 25 4 x- 3 y-27=0 & =400, =1728 & - =1328 aligned

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