# The line y = mx + c intersects the parabola la y^2= 4ax at the points P and Q.Show that the coordinates of the mid-point of PQ is ((2a-mc)/(m^2), (2a)/m) If the mid-point is M, find the locus of M whenm varies and c = 1

The line y = mx + c intersects theparabola ${y}^{2}=4ax$ at the points P and Q.Show that the coordinates of the mid-point of PQ is $\left(\frac{2a-mc}{{m}^{2}},\frac{2a}{m}\right)$.
If the mid-point is M, find the locus of M whenm varies and c = 1
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losnonamern
y=mx+c eqn (1)
${y}^{2}=ax$ eqn (2)
Square both side eqn 1
will get
${y}^{2}={m}^{2}{x}^{2}+2mcx+{c}^{2}$... (3)
set 2 and 3 equals
$4ax={m}^{2}{x}^{2}+2mcx+{c}^{2}$
or
$\left({m}^{2}\right){x}^{2}+\left(2mc-4a\right)x+\left({c}^{2}\right)=0$
$x=-\frac{b}{2a}=\frac{-\left(2mc-4a\right)}{2{m}^{2}}=\frac{2a-mc}{{m}^{2}}$
now plug in function 1
$y=m\ast \left(\frac{2a-mc}{{m}^{2}}\right)+c=\frac{2a-mc}{m}+\frac{cm}{m}=\frac{2a}{m}$
so we just showed that it works
$\left(\frac{2a-mc}{{m}^{2}},\frac{2a}{m}\right)$
next,
when c=1:
$\left(\frac{2a-m}{{m}^{2}},\frac{2a}{m}\right)$