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Linear Algebra Solutions 76

# Linear Algebra Solutions 76 - x-2 = x 1 y 1 = y 1 However...

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x 1 y 1 4.5 9 13.5 -4.5 -9 4.5 9 13.5 -4.5 -9 x y x 1 y 1 4 8 -4 -8 4 8 -4 -8 x y Figure 1: (a): x 2 - 8 x + 8 y + 8 = 0; (b): y 2 - 12 x + 2 y + 25 = 0 Section 7.3 1. (i) x 2 - 8 x +8 y +8 = ( x - 4) 2 +8( y - 1). So the equation x 2 - 8 x +8 y +8 = 0 becomes x 2 1 + 8 y 1 = 0 (1) if we make a translation of axes x - 4 = x 1 , y - 1 = y 1 . However equation (1) can be written as a standard form y 1 = - 1 8 x 2 1 , which represents a parabola with vertex at (4 , 1). (See Figure 1(a).) (ii) y 2 - 12 x + 2 y + 25 = ( y + 1) 2 - 12( x - 2). Hence y 2 - 12 x + 2 y + 25 = 0 becomes y 2 1 - 12 x 1 = 0 (2) if we make a translation of axes
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Unformatted text preview: x-2 = x 1 , y + 1 = y 1 . However equation (2) can be written as a standard form y 2 1 = 12 x 1 , which represents a parabola with vertex at (2 ,-1). (See Figure 1(b).) 2. 4 xy-3 y 2 = X t AX , where A = • 2 2-3 ‚ and X = • x y ‚ . The eigenvalues of A are the roots of λ 2 + 3 λ-4 = 0, namely λ 1 =-4 and λ 2 = 1. 79...
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