The image shows a graph of the parabola with focus (3,4) and directrix y=2, and the line give by y=4. Find a verify the points where the parabola and the line intersect
To find the points where the parabola and the line intersect, we can set the equations of the parabola and the line equal to each other and solve for x and y.
The equation of a parabola with focus (p, q) and directrix y = k can be written in the form:
(x - p)^2 = 4a(y - q)
Given that the focus is (3,4) and the directrix is y = 2, we can write the equation of the parabola as:
(x - 3)^2 = 8(y - 4)
The equation of the line is y = 4.
Substitute y = 4 into the equation of the parabola:
(x - 3)^2 = 8(4 - 4)
(x - 3)^2 = 0
Taking the square root of both sides:
x - 3 = 0
x = 3
So, the x-coordinate of the point of intersection is x = 3. Now substitute this value back into the equation of the line to find the y-coordinate:
y = 4
Therefore, the point of intersection is (3, 4).
To verify this solution, you can plot the equations of the parabola and the line on the graph. The point (3, 4) should lie on both the line and the parabola.