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. $\therefore$ ${\left(\frac{\lambda -6}{\lambda +1}\right)}^2=4\left(\frac{3\lambda +4}{\lambda +1}\right)$ $\Rightarrow$ $11{\lambda }^2+40\lambda -20=0$ $\Rightarrow$ $\lambda =\frac{-20\pm 2\sqrt{155}}{11}:1$