L is a straight line in x-y plane. its equation is in the form y-8=m(x-1)

determine all points in x-y plane where line L intersects curve y=2x^2

i know that 1,8 is a point on the line.

I'm not sure what points the question is looking for here. Without knowing what m is, every point except (1,2) is intercepted for some real m. You're correct, (1,8) is a point on the line, but that's not enough information to determine what points are intersected on the parabola.

To determine all the points where the line L intersects the curve y = 2x^2, we need to solve the system of equations formed by setting the equations of the line and the curve equal to each other.

First, let's substitute the equation of the line into the equation of the curve:

y - 8 = m(x - 1) (equation of the line)
y = 2x^2 (equation of the curve)

Substituting the value of y from the line equation into the curve equation, we get:

m(x - 1) - 8 = 2x^2

Rearranging the equation, we have:

2x^2 - mx + (8 - m) = 0

Now, to find the points of intersection, we need to solve this quadratic equation for x. Given that the quadratic equation is in the form ax^2 + bx + c = 0, we can find the solutions using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 2, b = -m, and c = 8 - m.

Using the quadratic formula, we can find the values of x where the line L intersects the curve y = 2x^2.

Please note that depending on the value of m, there may be zero, one, or two points of intersection.