First slide

Equation of line in Straight lines

Question

Let P be the point (-3, 0) and Q be a moving point (0,3t). Let PQ be trisected at R so that R is nearer to Q. RN is drawn perpendicular to PQ meeting the x-axis at N. The locus of the mid-point of RN is

Moderate

A

$(x + 3)^{2} - 3 y = 0$
B

$(y + 3)^{2} - 3 x = 0$
C

$x^{2} - y = 1$
D

$y^{2} - x = 1$

Solution

$S i n c e R i s p o i n t o f t r i s e c t i o n R \Rightarrow R d i v i d e s P Q i n t h e r a t i o 2 : 1 = (\frac{2 (0) + 1 (- 3)}{3}, \frac{2 (3 t) + 1 (0)}{3})$
$P (- 3, 0), Q (0, 3 t), R (- 1, 2 t)$
$Let the mid point of R N be (h, k)$
$\frac{R (- 1, 2 t) + N (x, y)}{2} = (h, k)$
$- 1 + x = 2 h 2 t + y = 2 k$
N=(2h+1,0)
$\begin{array}{ll} ∴ k = t \\ R N ⊥ P Q \Rightarrow \frac{2 t}{- 2 h - 2} \times \frac{3 t}{3} = - 1 \\ \Rightarrow 2 t^{2} = 2 h + 2 \\ \Rightarrow t^{2} = h + 1 \\ \Rightarrow k^{2} = h + 1 \\ Locus of (h, k) is y^{2} = x + 1 \end{array}$

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