# Equation of a locus

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• Sep 5th 2009, 02:00 AM
saberteeth
Equation of a locus
(Hi) there,
I was unable to solve the following questions, please help me.

1) A is a point on the X-axis and B is a point on the Y-axis 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 $\displaystyle y^2 = 12x$
• Sep 5th 2009, 08:58 AM
red_dog
1) $\displaystyle A(a,0), \ B(0,b), \ P\left(\frac{a}{2},\frac{b}{2}\right)$

$\displaystyle 4OA+7OB=20\Rightarrow 4|a|+7|b|=20$.

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 $\displaystyle 4|x|+7|y|=10$

The locus is a parallelogram formed by the lines:

$\displaystyle 4x+7y=10$

$\displaystyle -4x-7y=10$

$\displaystyle 4x-7y=10$

$\displaystyle -4x+7y=10$

2) $\displaystyle A(-1,2), Q\left(\frac{a^2}{12},a\right), \ P\left(\frac{a^2-12}{24},\frac{a+2}{2}\right)$

Then $\displaystyle x=\frac{a^2-12}{24}, \ y=\frac{a+2}{2}$

$\displaystyle a=2y-2\Rightarrow x=\frac{y^2-2y+2}{6}$.

The locus is a parabola.