Example 5: Same of Hyperbolic Paraboloid

Find the tangent line and slope of the level curve of the hyperbolic paraboloid

z = x² - y²

at the point (a, b) (where b ≠ 0) (Figure 11.6.7).

Figure 11.6.7: Level curves of z = x² - y²

The level curve has the equation

x² - y² = a² - b², x² - y² - (a² - b²) = 0.

Put

w = x² - y² - (a² - b²) = 0.

Then

At (a, b),

Tangent Line:

2a(x - a) - 2b(y - b) = 0.

Slope: