A contour at height c have [tex]\mathbf{f(x, y) = c}[/tex] as its equation
The equation of the contour is [tex]\mathbf{2x^2y + 7x + 15 = 0}[/tex]
The given parameters are:
[tex]\mathbf{f(x,y) = 2x^2y + 7x + 20}[/tex]
[tex]\mathbf{(x,y) = (3,-2)}[/tex]
Recall that:
[tex]\mathbf{f(x, y) = c}[/tex]
Substitute [tex]\mathbf{(x,y) = (3,-2)}[/tex] in f(x,y)
[tex]\mathbf{f(x,y) = 2x^2y + 7x + 20}[/tex]
[tex]\mathbf{f(3,-2) = 2 \times 3^2 \times -2 + 7 \times 3 + 20}[/tex]
Evaluate exponents
[tex]\mathbf{f(3,-2) = 2 \times 9 \times -2 + 7 \times 3 + 20}[/tex]
Evaluate the products
[tex]\mathbf{f(3,-2) = -36 + 21 + 20}[/tex]
[tex]\mathbf{f(3,-2) = 5}[/tex]
Replace 3 and -2, with x and y
[tex]\mathbf{f(x,y) = 5}[/tex]
Substitute 5 for f(x,y) in [tex]\mathbf{f(x,y) = 2x^2y + 7x + 20}[/tex]
[tex]\mathbf{2x^2y + 7x + 20 = 5}[/tex]
Collect like terms
[tex]\mathbf{2x^2y + 7x + 20 - 5 = 0}[/tex]
[tex]\mathbf{2x^2y + 7x + 15 = 0}[/tex]
Hence, the equation of contour is [tex]\mathbf{2x^2y + 7x + 15 = 0}[/tex]
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