
Equation of a locus
(Hi) there,
I was unable to solve the following questions, please help me.
1) A is a point on the Xaxis and B is a point on the Yaxis such that:
4(OA) + 7(OB) = 20, where O is the origin. Find the equation of the locus of the midpoint P of Segment AB.
2) Find the equation of the locus of the midpoint P of Segment AQ, if:
A= (1,2) and Q is a point on the locus $\displaystyle y^2 = 12x$

1) $\displaystyle A(a,0), \ B(0,b), \ P\left(\frac{a}{2},\frac{b}{2}\right)$
$\displaystyle 4OA+7OB=20\Rightarrow 4a+7b=20$.
Divide by 2:
$\displaystyle 4\left\frac{a}{2}\right+7\left\frac{b}{2}\right =10$
Then the coordinates of P verify the equation $\displaystyle 4x+7y=10$
The locus is a parallelogram formed by the lines:
$\displaystyle 4x+7y=10$
$\displaystyle 4x7y=10$
$\displaystyle 4x7y=10$
$\displaystyle 4x+7y=10$
2) $\displaystyle A(1,2), Q\left(\frac{a^2}{12},a\right), \ P\left(\frac{a^212}{24},\frac{a+2}{2}\right)$
Then $\displaystyle x=\frac{a^212}{24}, \ y=\frac{a+2}{2}$
$\displaystyle a=2y2\Rightarrow x=\frac{y^22y+2}{6}$.
The locus is a parabola.