What is the smallest distance between the origin and a point on the graph of y = 1/2*(x^2 - 8)?
+289 y = 1/2x^2 - 4
Suppose that \((x_0, y_0)\) is a point on this graph closest to the origin.
Hence:
\(y_0 = \frac12(x_0)^2 - 4.\)
Using this information, we need to find
\(\sqrt{x_0^2 + y_0^2}\)
We can plug in \(x_0^2\) with \(2y_0 + 8\) (Make sure you see how I got this)
Hence, the distance between the points will be:
\(\sqrt(y_0^2 +2y_0 + 8)\)
Since this distance is to be minimized by how we defined the point (x_0, y_0), we can find the vertex of the that quadratic in y_0. If you complete the square, you will recieve that the parabola reaches its minimum when \(y_0 = -1\).
Can you solve the problem from here?
