Let P be the point (1, 0) and Q a point on the parabola y^2 = 8x.

Question

Let $P$ be the point $(1, 0)$ and $Q$ a point on the parabola ${y^2} = 8x$. The locus of mid point of $PQ$ is :

Accepted Answer

$$P = \left( {1,0} \right)\,\,Q = \left( {h,k} \right)$$ Such that ${k^2} = 8h$

Let $$\left( {\alpha ,\beta } \right)$$ be the midpoint of $PQ$

$$\alpha  = {{h + 1} \over 2},\,\,\,\beta  = {{k + 0} \over 2}$$

$$ \therefore $$ $$2\alpha  - 1 = h\,\,\,\,\,\,2\beta  = k.$$

$${\left( {2\beta } \right)^2} = 8\left( {2\alpha  - 1} \right) \Rightarrow {\beta ^2} = 4\alpha  - 2$$

$$ \Rightarrow {y^2} - 4x + 2 = 0.$$

Solutions · 1