A normal with slope 1 6 is drawn from the point (0,- ) to the parabola x^2=-4 a y, where a>0. Let L be the line passing through (0,- ) and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B. Let r denote the length of the latus rectum and s denote the square of the length of the line segment A B. If r: s=1: 16, then the value of 24ais ________.
A. A
d y d x = . x -2 a d y d x |_N=-t Slope of normal = 1 t = 1 6 t= 6 Now, - at ^2+ 2 at = 1 t - at ^2+ =2 a -6 a + =2 a =8 a For A and B aligned & x^2=-4 a(-8 a) \\ & x^2=32 a^2 x= 4 2 a \\ & A(-4 2 a,-8 a), B(4 2 a,-8 a) \\ & A B^2=(8 2 a)^2=128 a^2=s aligned Length of LR = r =4 a aligned & r s = 4 a 128 a ^2 = 1 16 \\ & 32 a =16 a = 1 2 \\ & 24 a =12 Ans. aligned
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