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
The equation of the variable line is :
But we know that it's passing through (5,0).
Therefore, the equation of the variable line is : , a is the variable
So we know that :
Because H and K are on this line.
Plus, H is on the line of equation
And K is on the line of equation
By substituting (1) into (3), you will have with respect to a.
Replace this expression of in (1) to have with respect to a.
Do the same for and .
Then, the midpoint of HK is , which will be coordinates with respect to a
Hope that helps... If you have questions about some steps, don't hesitate ^^