the graph shows a circle with a radius 1 inscribed in the parabola y=x^2. Find the center of the circle.
To find the center of the circle inscribed in the parabola, we need to find the point where the parabola intersects the circle.
The equation of a circle with center (h, k) and radius r is given by the equation:
(x - h)^2 + (y - k)^2 = r^2
In this case, the circle has a radius of 1. So, the equation of the circle would be:
(x - h)^2 + (y - k)^2 = 1
Now, let's substitute the equation of the parabola into the equation of the circle to find the intersection points.
Substituting the y-coordinate of the parabola (y = x^2) into the equation of the circle:
(x - h)^2 + (x^2 - k)^2 = 1
Expanding the equation:
(x^2 - 2hx + h^2) + (x^4 - 2kx^2 + k^2) = 1
Rearranging and combining similar terms:
x^4 + (1 - 2k)x^2 + (h^2 - 2hx + k^2 - 1) = 0
For the circle to intersect the parabola at one point, these two curves should be tangent to each other. This means that the quadratic equation obtained above should have only one root.
In a quadratic equation, the discriminant (b^2 - 4ac) determines the number of roots:
Discriminant = (1 - 2k)^2 - 4(1)(h^2 - 2hx + k^2 - 1) = 0
Expanding and simplifying:
1 - 4k + 4k^2 - 4h^2 + 8hx - 4k^2 + 4 = 0
Simplifying further:
8hx - 4h^2 - 4k + 5 = 0
This equation relates the values of h and k for the circle to be tangent to the parabola. By solving this equation, you can find the coordinates of the center of the circle (h, k).