Final answer:
To find the intersection points between a curve and a paraboloid, set their equations equal and solve the resulting system of equations for x, y, and z. Substitute the x-coordinates back into the curve equation to find the y-coordinates. Substitute both the x and y coordinates into the paraboloid equation to find the z-coordinates.
Step-by-step explanation:
The curve intersects the paraboloid at certain points where they share common coordinates. To find the intersection points, you need to set the equation of the curve equal to the equation of the paraboloid. Then solve the resulting system of equations to find the values of x, y, and z at the intersection points.
For example, if the equation of the curve is y = x^2 and the equation of the paraboloid is z = x^2 + y^2, you can substitute y = x^2 into the equation of the paraboloid to get z = x^2 + (x^2)^2. Simplify this equation to get z = x^2 + x^4. Now you can set this equation equal to a constant to find the intersection points.
For instance, if you set z = x^2 + x^4 equal to 0, you can use algebraic techniques or a graphing calculator to solve this equation and find the x-coordinates of the intersection points. Then substitute these x-coordinates back into the original curve equation to find the corresponding y-coordinates. Finally, substitute both the x and y coordinates back into the equation of the paraboloid to find the z-coordinates of the intersection points.