Answer:
Step-by-step explanation:
To find the coordinates of the points where the line and the surface intersect, we can substitute the equations of the line into the equation of the surface and solve for t. We are looking for values of t that satisfy both equations, which will give us the coordinates of the points of intersection.
Substituting the equations of the line into the equation of the surface, we get:
x^2 + y^2 = (t)^2 + (1+t)^2 = 2t^2 + 2t + 1
z = 5t
We can now substitute the expression for z into the first equation:
x^2 + y^2 = 2t^2 + 2t + 1
Since we know that z = x^2 + y^2, we can substitute this expression for z into the equation above:
z = 2t^2 + 2t + 1
Setting these two expressions for z equal to each other and solving for t, we get:
5t = 2t^2 + 2t + 1
2t^2 - 3t + 1 = 0
This quadratic equation can be factored as:
(2t - 1)(t - 1) = 0
So the solutions are t = 1/2 and t = 1.
For t = 1/2, the coordinates of the point of intersection are:
x = t = 1/2
y = 1 + t = 3/2
z = 5t = 5/2
Therefore, one point of intersection is (1/2, 3/2, 5/2).
For t = 1, the coordinates of the point of intersection are:
x = t = 1
y = 1 + t = 2
z = 5t = 5
Therefore, the other point of intersection is (1, 2, 5).
Hence, the coordinates of the two points of intersection between the line and the surface are (1/2, 3/2, 5/2) and (1, 2, 5).