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Distance from a plane to an ellipsoid (Adapted from 1938
Chapter 10, Problem 62AE(choose chapter or problem)
Distance from a plane to an ellipsoid (Adapted from 1938 Putnam Exam.) Consider the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) and the plane P given by Ax + By + C2 + 1 = 0. Let \(h=\left(A^{2}+B^{2}+C^{2}\right)^{-1 / 2}) and\(m=\left(a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}\right)^{1 / 2} .\)
a. Find the equation of the plane tangent to the ellipsoid at the point (p, q, r).
b. Find the two points on the ellipsoid at which the tangent plane is parallel to P and find equations of the tangent planes.
c. Show that the distance between the origin and the plane P is h.
Questions & Answers
QUESTION:
Distance from a plane to an ellipsoid (Adapted from 1938 Putnam Exam.) Consider the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) and the plane P given by Ax + By + C2 + 1 = 0. Let \(h=\left(A^{2}+B^{2}+C^{2}\right)^{-1 / 2}) and\(m=\left(a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}\right)^{1 / 2} .\)
a. Find the equation of the plane tangent to the ellipsoid at the point (p, q, r).
b. Find the two points on the ellipsoid at which the tangent plane is parallel to P and find equations of the tangent planes.
c. Show that the distance between the origin and the plane P is h.
ANSWER:Solution 62AEStep 1 of 5:Given :The equation of the ellipsoid is . The plane (P) is : , and also given by h = , m = 1. In this problem we need to find the equation of the plane tangent to the ellipsoid at ( p , q , r) Let us consider , F(x , y, z) : So , FORMULA: An equation of the tangent plane to the surface z = f(x , y , z) at the point P(is : ……….(1)Here , ) denotes the partial derivative of ‘f’ with respect to x at , ) denotes the partial derivative of ‘f’ with respect to y at and ) denotes the partial derivative of ‘f’ with respect to z at .Now , the partial derivatives are : , since , here y and z ‘s are constants .Therefore , .Similarly , and …………….(2)At the point ( p , q , r) the values of : Therefore , the equation of the tangent plane is : , since from (1). …………(3)