Question

31. Let Q be the point on an ellipse closest to a given point P outside the ellipse. It was known to the Greek mathematician Apollonius (third century BCE) that PQ is perpendicular to the tangent to the ellipse at Q (Figure 16). Explain in words why this conclusion is a consequence of the method of Lagrange multipliers. Hint: The circles centered at P are level curves of the function to be minimized. Rogawski et al., Multivariable Calculus, 4e, © 2019 W. H. Freeman and Company FIGURE 16

Answer 1

.P(X, Yo) 2 x² Let P(xo, yo) be the point. We want the coordinates of a (x,y) such that Fesy) = |PQ TE (-xHy-Ya! is minimum. But Q lies on the ellipse say az + 1. So the optimisation problem is : Minimise : f (xy) = 60–260) + (y-y)* subject to Suppose the minimum value of flogy) = k then (e-20)*+ly-y)=k + bcy) lies on a drce with center p l radius pé). So this arde meets the ellipse at a. But then the tangent of the circle te ellibee at eis same le we knows that radius of circle is I to tangent at any point. I hus på is I the tangent at B. D + -=0. a