2007 ToT Spring Seniors A1: Coordinate Bashing

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.

Start out by graphing y=x^2, and labeling lines \overline{AB} and \overline{CD}, such that all points A,B,C,D are on the line y=x^2. We label the point A (a,a^2), the point B (b,b^2), the point C (c,c^2), and the point D(d,d^2). The lines intersect at the y-axis, which we label to be point (0,e). 

We start by finding the equation \overline{AB} is on. We put the line into the slope-intercept form y=mx+z, where m is the slope and z is the y-intercept* . Since the line intersects the y-axis at the point (0,e), we know the y-intercept is e. The slope of the line is \frac{b^2-a^2}{b-a}=\frac{(b+a)(b-a)}{(b-a)}=b+a. Therefore we have y=(a+b)\times x+e. Substituting the point (a,a^2) into the equation gives us a^2=(a+b)\times a+e\implies a^2=a^2+ab+e\implies ab+e=0.
We could also substitute (b,b^2) into the equation to give us b^2=(a+b)\times b+e\implies ab+e=0 as well.

We now find the equation \overline{CD} is on. Again, put the line into slope-intercept form y=mx+z, where m is slope, and z is y-intercept*. The lines \overline{AB} and\overline{CD} intersect at the point (0,e)\forall e, therefore the y-intercept of \overline{CD} is e. The slope of \overline{CD} is \frac{d^2-c^2}{d-c}=\frac{\left(d+c\right)\left(d-c\right)}{\left(d-c\right)}= d+c. Therefore our new equation is y= \left(d+c\right)\times x+e. Substituting either points (c,c^2) or (d,d^2) give c^2= \left(d+c\right)\times c+e or d^2= \left(d+c\right)\times d+e, where both can be simplified two after expanding and subtract c^2 in the first equation, and d^2 in the second equation to give cd+e=0. 

We are now left with the two equations cd+e=0 and ab+e=0. Subtracting the two equations results in 
\implies cd=ab
\implies d=\boxed{\frac{ab}{c}}

*(note: I changed the slope-intercept form from y=mx+b to y=mx+z to avoid any confusion with point B)


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