Given the quadratic
`Q(x) = x^2 - 2x_0x + 4ay_0`
suppose the points A, B lie on a line tangential to Q(x) and that the x-coordinates of A and B are equal to the x-coordinates of the roots of Q(x).
Find the coordinates of the points A and B given that the line tangential to Q(x) touches Q(x) at `x=x_1`.
First find the x-coordinates of A and B respectively, which are equal to the two roots of the quadratic Q(x). To find these roots `x(A)`, `x(B)` we solve
`Q(x) = 0`
Using the quadratic formula `x = (-beta pm sqrt(beta^2-4alpha gamma))/(2 alpha)` where we write
`Q(x)` as `alpha x^2 + beta x + gamma` we then have that
`x(A)`, `x(B)` `= (2x_0 pm sqrt(4x_0^2 - 16ay_0))/2` `= x_0 pm sqrt(x_0^2 - 4ay_0)`
Now, the tangent line to Q(x) on which A and B lie has gradient equal to the gradient function of Q(x), Q'(x). We obtain this gradient function by differentiating Q(x) with respect to x:
`Q'(x) = 2(x - x_0)`
Since the tangent line touches Q(x) at (x,Q(x)) we know that that point is on the tangent line.
Consider a point `x_1` where a particular tangent line `T_1(x)` touches the parabola Q(x). ``The gradient of `T_1(x)` is the gradient of Q(x) at `x_1`, `2(x_1-x_0)`. `T_1(x)` also passes through the point (`x_1, Q(x_1))` so that` `
`T_1(x) = 2(x_1-x_0)x + Q(x_1) - 2(x_1-x_0)x_1`
`= 2(x_1-x_0)(x-x_1) + Q(x_1)`.
Since this is the equation for the tangent line on which A and B lie then the y-coordinates of A and B are given by `T_1(x(A))` and `T_1(x(B))` respectively.
Note that if `x_0^2 < 4ay_0` (the discriminant of Q(x) is < 0 ), then A and B do not exist as the parabola doesn't cross the x-axis at all. If `x_0^2 = 4ay_0` then A=B as the parabola touches the x-axis at A=B.
A has coordinates (`x_0 - sqrt(x_0^2 - 4ay_0)` , `T_1(x_0-sqrt(x_0^2-4ay_0))`)
B has coordinates (`x_0 + sqrt(x_0^2 - 4ay_0)`, `T_1(x_0+sqrt(x_0^2-4ay_0))`)
The x-coordinates remain the same and the y-coordinates depend on `(x_0,y_0)` and the point the tangent `T_1(x)` touches Q(x), `x_1`.