p311 q12
question : a variable line passing through the point (5,0) intersects the lines 3x-4y=0 and 3x+4y=0 at H and K respectively. Find the equation of the locus of the mid-point of HK.
thanks in advance
Hello,
Let $\displaystyle H(x_H, y_H)$ and $\displaystyle K(x_K, y_K)$
The equation of the variable line is : $\displaystyle y=ax+b$
But we know that it's passing through (5,0).
Thus $\displaystyle 0=5a+b$
--> $\displaystyle b=-5a$
Therefore, the equation of the variable line is : $\displaystyle \boxed{y=ax-5a}$, a is the variable
So we know that :
$\displaystyle \begin{aligned} y_H & = & ax_H-5a & (1) \\ y_K & = & ax_K-5a & (2) \end{aligned}$
Because H and K are on this line.
Plus, H is on the line of equation $\displaystyle 3x-4y=0$
And K is on the line of equation $\displaystyle 3x+4y=0$
---> $\displaystyle \begin{aligned} 3x_H-4y_H & = & 0 & (3) \\ 3x_K+4y_K & = & 0 & (4) \end{aligned}$
By substituting (1) into (3), you will have $\displaystyle x_H$ with respect to a.
Replace this expression of $\displaystyle x_H$ in (1) to have $\displaystyle y_H$ with respect to a.
Do the same for $\displaystyle x_K$ and $\displaystyle y_k$.
Then, the midpoint of HK is $\displaystyle M \left(\frac{x_H+x_K}{2}, \frac{y_K+y_K}{2} \right)$, which will be coordinates with respect to a
Hope that helps... If you have questions about some steps, don't hesitate ^^