# Let P (x, y) be the midpoint of the line joining (1, 0) to the point on the curve y^2= (x+1)(x+5) – (x+2)(x+3). The locus of P is symmetric about ?

## a). y-axis b). x=1 c). y=1 d). x-axis e). (0, 0)

Dec 13, 2016

See below.

#### Explanation:

Note that ${y}^{2} = \left(x + 1\right) \left(x + 5\right) - \left(x + 2\right) \left(x + 3\right) = x - 1$

so the curve is the parabola

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

The mid point locus $P \left(x , y\right) = \frac{1}{2} \left(x + 1 , y\right)$ is symmetric about the $x$ axis

See attached plot.

In red ${y}^{2} = x - 1$ and in blue $P \left(x , y\right) = \frac{1}{2} \left(x + 1 , y\right)$