Suppose C is the elliptic curve y^2 − x^3 − a = 0 where "a" is a rational number. Suppose P = (p, q) is a point on C with rational coordinates. Suppose further that L is the tangent line to C at point P. Let Q ≠ P be an intersection point of C and L. Then Q has rational coordinates.
To prove that Q has rational coordinates, we need to show that the intersection point of the curve C and the tangent line L has rational coordinates. We can do this by using the equation of the curve and the equation of the tangent line.
First, let's find the equation of the tangent line L at point P. For a point (x, y) on the curve C, the slope of the tangent line L is given by the derivative of the curve equation with respect to x, evaluated at the point P. Let's denote this slope as m.
The derivative of the curve equation y^2 - x^3 - a = 0 with respect to x is given by:
dy/dx = (3x^2) / (2y)
Now, substitute the coordinates of the point P = (p, q) into the derivative:
dy/dx = (3p^2) / (2q)
Since the tangent line L has slope m, we can write its equation in point-slope form:
y - q = m(x - p)
Substituting the slope m, we get:
y - q = ((3p^2) / (2q))(x - p)
This is the equation of the tangent line L at point P.
Now, let's find the coordinates of the intersection point Q = (r, s), which lies on the curve C and the tangent line L. We can substitute the coordinates of Q into both the curve equation and the equation of the tangent line and solve for r and s.
Substituting the coordinates of Q into the curve equation y^2 - x^3 - a = 0 gives:
s^2 - r^3 - a = 0
Substituting the coordinates of Q into the equation of the tangent line y - q = ((3p^2) / (2q))(x - p) gives:
s - q = ((3p^2) / (2q))(r - p)
Now, we have two equations:
1. s^2 - r^3 - a = 0
2. s - q = ((3p^2) / (2q))(r - p)
To show that Q has rational coordinates, we need to find a rational solution for r and s satisfying both equations.
Solving these equations generally involves identifying a rational solution by making some assumptions or using additional properties of elliptic curves. However, based on the given information, we cannot guarantee a general rational solution for Q without knowing more about the specific values of "a," "p," and "q" on the elliptic curve.
In conclusion, while it is possible to find rational coordinates for the intersection point Q on the curve C and the tangent line at point P, it requires additional information or assumptions about the values of "a," "p," and "q" to provide a definite answer.