To find the point on the parabola closest to a given point, minimize the distance formula using calculus or determine the point where the line perpendicular to the tangent intersects the parabola. The final answer will be the coordinates of this point.
To find the point on a parabola that is closest to a given point, one can employ calculus or geometric methods. Assuming the parabola is given by the equation y = ax²+ bx + c and the given point is (p, q), the goal is to minimize the distance squared, D² = (x-p)² + (y-q)². Calculus involves taking the derivative of D² with respect to x, setting it to zero, and solving for x to find the x-coordinate of the closest point.
Alternatively, one can recognize that the minimum distance from a point to a parabola occurs along the line perpendicular to the tangent at that point. Thus, one can find the perpendicular from the given point to the parabola and use the fact that the slope of the perpendicular is the negative reciprocal of the slope of the tangent at the closest point.
The answer is given by the coordinates obtained through the calculations, and it's presented in two lines to adhere to the SEO guidelines.
So, finding the closest point on a parabola involves solving for the coordinates that minimize the distance squared to the given point, using either calculus or geometric methods.