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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let $ A $ be the focus of the parabola $ y^2 = 8x $. Let the line $ y = mx + c $ intersect the parabola at two distinct points $ B $ and $ C $. If the centroid of the triangle $ ABC $ is $ \left( \frac{7}{3}, \frac{4}{3} \right) $, then $ (BC)^2 $ is equal to :
Solution
Coordinates of centroid of triangle $ ABC $ are
$ \frac{2}{3}(t_1^2 + t_2^2 + 1) = \frac{7}{3} \Rightarrow t_1^2 + t_2^2 = \frac{5}{2} $
$ \frac{4}{3}(t_1 + t_2) = \frac{4}{3} \Rightarrow t_1 + t_2 = 1 $
$ (t_1 + t_2)^2 = t_1^2 + t_2^2 + 2t_1 t_2 \Rightarrow t_1 t_2 = -\frac{3}{4} $
$ (t_1 - t_2)^2 = (t_1 + t_2)^2 - 4t_1 t_2 = 4 $
$ (BC)^2 = 4(t_1 - t_2)^2 + 16(t_1 - t_2)^2 $
$ \Rightarrow (BC)^2 = 80 $
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