First slide

Parabola

Question

A point R divides the line segment joining the points $P (1, 3)$ and $Q (1, 1)$ in the ratio $1 : λ$ If R is an interior point of the parabola $y^{2} = 4 x,$ then $λ$ can take a value in the interval

Moderate

A

$- 1 < λ < \frac{3}{5}$
B

$- \frac{3}{5} < λ < 5$
C

$- 2 < λ < 2$
D

$- \frac{3}{5} < λ < 1$

Solution

Coordinates of R are
$\begin{array}{l} = (\frac{λ + 1}{λ + 1}, \frac{3 λ + 1}{λ + 1}) \\ = (1, \frac{3 λ + 1}{λ + 1}) \end{array}$
$\Rightarrow$ For all values of $λ$ x-coordinate of R is 1 and y-coordinate of R varies between –2 and 2
$\begin{array}{l} \Rightarrow - 2 < \frac{3 λ + 1}{λ + 1} < 2 . \\ \Rightarrow - \frac{3}{5} < λ < 1 \end{array}$

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