Find the point on the parabola x+y^2=0 that is closest to the point (0,-3).
To find the point on the parabola x + y^2 = 0 that is closest to the point (0, -3), we need to minimize the distance between these two points.
Let's call the coordinates of the point on the parabola that is closest to (0, -3) as (a, b).
The distance between two points (x1, y1) and (x2, y2) is given by the formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
So, in this case, the distance between (0, -3) and (a, b) is:
Distance = √[(a - 0)^2 + (b - (-3))^2]
= √[a^2 + (b + 3)^2]
Now, we need to minimize this distance. To do that, we can minimize the square of the distance, as the square root is a monotonically increasing function (meaning that if A < B, then √A < √B).
So, we can minimize the square of the distance:
Distance^2 = a^2 + (b + 3)^2
We need to find the values of a and b that minimize this expression, while also satisfying the equation of the parabola x + y^2 = 0.
To find the minimum of Distance^2, we can take partial derivatives of Distance^2 with respect to a and b, and set them equal to zero. This will give us a system of equations that we can solve to find the values of a and b.
Differentiating Distance^2 with respect to a:
∂(Distance^2) / ∂a = 2a + 0 = 0
So, 2a = 0
Therefore, a = 0
Differentiating Distance^2 with respect to b:
∂(Distance^2) / ∂b = 0 + 2(b + 3) = 0
So, 2(b + 3) = 0
Therefore, b + 3 = 0
Thus, b = -3
So, we have found that a = 0 and b = -3, which means the point on the parabola x + y^2 = 0 that is closest to (0, -3) is (0, -3).
Therefore, the point (0, -3) itself is the closest point on the parabola to the point (0, -3).