Question

The ends of a line segment are $P Q : Q R = 1 : \lambda$. If R is an interior point of the parabola $y ^ { 2 } = 4 x$, then

• A. $\lambda \in ( 0,1 )$
• B. $\lambda \in \left( - \frac { 3 } { 5 } , 1 \right)$
• C. $\lambda \in \left( \frac { 1 } { 2 } , \frac { 3 } { 5 } \right)$
• D. None of these

Answer 1

Answer: (A)

$\lambda \in ( 0,1 )$

Answer 2

$R = \left( 1 , \frac { 1 + 3 \lambda } { 1 + \lambda } \right)$ It is an interior point of $y ^ { 2 } - 4 x = 0$

if $\left( \frac { 1 + 3 \lambda } { 1 + \lambda } \right) ^ { 2 } - 4 < 0$

⇒ $\left( \frac { 1 + 3 \lambda } { 1 + \lambda } - 2 \right) \left( \frac { 1 + 3 \lambda } { 1 + \lambda } + 2 \right) < 0$ ⇒$\left( \frac { \lambda - 1 } { 1 + \lambda } \right) \left( \frac { 5 \lambda + 3 } { 1 + \lambda } \right) < 0$

⇒ $( \lambda - 1 ) \left( \lambda + \frac { 3 } { 5 } \right) < 0$

Therefore, $- \frac { 3 } { 5 } < \lambda < 1$ . But $\lambda > 0$ $\therefore 0 < \lambda < 1 \Rightarrow \lambda \in ( 0,1 )$