Let S be the hyperboloid x 2 + y 2 = z2 + 1 and let P = (a, {3, 0) be a point on S in the (x, y )-plane. Show that there are precisely two lines through P entirely contained in S (Figure 18). Hint: Consider the line r(t) = (a +at, f3 + bt, t) through P. Show that r(t) is contained in S if (a, b) is one of the two points on the unit circle obtained by rotating (a, {3) through ± ~. This proves that a hyperboloid of one sheet is a doubly ruled surface, which means that it can be swept out by moving a line in space in two different ways.
Let S be the hyperboloid x 2 + y 2 = z2 + 1 and let P = (a, {3, 0) be a point on S in the (x, y )-plane. Show that there are precisely two lines through P entirely contained in S (Figure 18). Hint: Consider the line r(t) = (a +at, f3 + bt, t) through P. Show that r(t) is contained in S if (a, b) is one of the two points on the unit circle obtained by rotating (a, {3) through ± ~. This proves that a hyperboloid of one sheet is a doubly ruled surface, which means that it can be swept out by moving a line in space in two different ways.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 13AEXP
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Let S be the hyperboloid x 2 + y 2 = z2 + 1 and let P = (a, {3, 0) be a point on S in the (x, y )-plane. Show that there are precisely two lines through P entirely contained in S (Figure 18). Hint: Consider the line r(t) = (a +at, f3 + bt, t) through P. Show that r(t) is contained in S if (a, b) is one of the two points on the unit circle obtained by rotating (a, {3) through ± ~. This proves that a hyperboloid of one sheet is a doubly ruled surface, which means that it can be swept out by moving a line in space in two different ways.
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