Question

The ratio in which the line segment joining the points $ (4,-6) $ and (3, 1) is divided by the parabola $ {{y}^{2}}=4x $ is

• A. $ \frac{-20\pm \sqrt{155}}{11}:1 $
• B. $ \frac{-2\pm 2\sqrt{155}}{11}:2 $
• C. $ -20\pm 2\sqrt{155}:11 $
• D. $ -20\pm \sqrt{155}:11 $

Answer 1

Answer: (C)

Let the point $ P(h,k) $ on the parabola divides the line joining $ A(4,-6) $ and B (3,1) in ratio $ \lambda :1 $ Then, we have $ P(h,k)=\left( \frac{3\lambda +4}{\lambda +1},\frac{\lambda -6}{\lambda +1} \right) $ This point lies on the parabola. $ \therefore $ $ {{\left( \frac{\lambda -6}{\lambda +1} \right)}^{2}}=4\left( \frac{3\lambda +4}{\lambda +1} \right) $ $ \Rightarrow $ $ 11{{\lambda }^{2}}+40\lambda -20=0 $ $ \Rightarrow $ $ \lambda =\frac{-20\pm 2\sqrt{155}}{11}:1 $