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 midpoint of RN is
(a) ${(x+3)}^2-3y=0$
(b) $${(y+3)}^2-3x=0$$
(c) $$x^2-y=1$$
(d) $$y^2-x=1$$

Answer 1

Hint: Point of trisection means a point which exactly divides a line segment into three equal parts. Here, point R divides a line segment PQ into 3 parts but point R is closer to Q hence dividing the line segment PQ in a ratio of 2:1 internally. First we have to find out the point of division and then the locus of midpoint by applying the midpoint formula.

Complete step-by-step answer:

 The point P which divides the line segment joining the points A $$\left(x_{1,}{y}_1\right)$$, B $$(x_2,y_2)$$in the ratio m:n internally is given by
P = $$({\displaystyle \frac{m{x}_2+n{x}_1}{m+n}},{\displaystyle \frac{m{y}_2+n{y}_1}{m+n}})$$
P(3,0) and Q(0,3t) are given. Now we have to find R which divides P and Q internally in the ratio 2:1.
R= $$({\displaystyle \frac{2(0)+1(-3)}{2+1}},{\displaystyle \frac{2(3t)+1(0)}{2+1}})$$
R = (-1,2t)
Now consider the point N as (x,0) because it is lying on the x axis and the line RN is perpendicular to PQ.
Slope of PQ $\times$slope of RN = -1
Because the product of slopes of two perpendicular lines is -1.
$${\displaystyle \frac{3t-0}{0-(-3)}}\times {\displaystyle \frac{2t-0}{-1-x}}=-1$$
$\begin{array}{rl}& 2t^2=x+1\\ & x=2t^2-1\end{array}$
$\begin{array}{rl}& 2t^2=x+1\\ & x=2t^2-1\end{array}$
 The point N is $$(2t^2-1,0)$$

 The locus of midpoint of RN is
Formula for midpoint is: $$({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{y_1+y_2}{2}})$$
The midpoint of RN is $$({\displaystyle \frac{2t^2-1+(-1)}{2}},{\displaystyle \frac{2t+0}{2}})$$
(x,y) = $$(t^2-1,t)$$
Therefore y=t, x= $$t^2-1$$
$$\therefore$$ $$x=y^2-1$$
Therefore the option is d.

So, the correct answer is “Option d”.

Note: When two lines are perpendicular to each other the product of their slopes is equal to -1. R can divide P and Q in the ratio 1:2 and 2:1 internally. Here we have to take 2:1 because it was given that R is closer to Q.