mathforthought

$A,B,C$ and $D$ are points on the parabola $y = x^2$ such that $AB$ and $CD$ intersect on the $y$ -axis. Determine the $x$ -coordinate of $D$ in terms of the $x$ -coordinates of $A,B$ and $C$ , which are $a, b$ and $c$ respectively.

Solution:


Start out by graphing , and labeling lines  and , such that all points  are on the line . We label the point , the point , the point , and the point . The lines intersect at the y-axis, which we label to be point . 

We start by finding the equation  is on. We put the line into the slope-intercept form , where  is the slope and  is the y-intercept* . Since the line intersects the  at the point , we know the y-intercept is . The slope of the line is . Therefore we have . Substituting the point  into the equation gives us .
We could also substitute  into the equation to give us  as well.

We now find the equation  is on. Again, put the line into slope-intercept form , where  is slope, and  is y-intercept*. The lines  and intersect at the point , therefore the y-intercept of  is . The slope of  is . Therefore our new equation is . Substituting either points  or  give  or , where both can be simplified two after expanding and subtract  in the first equation, and  in the second equation to give . 

We are now left with the two equations  and . Subtracting the two equations results in 




*(note: I changed the slope-intercept form from  to  to avoid any confusion with point )