start with the hyperbolic paraboloid surface z fx y y2 x2 and an arbitrary point pa b c on the s

Hi. Now point A, B, C lies on the hyperbolic paraboloid, Z is equal to Y square minus X square.

So being given the point A, B, C with an equation of Z equals Y squared minus X squared, we can

Okay, so a parameterization here is very easy to find and it's just x comma y, comma x squared p

We're asked to show that if a point A, E, C lies on the hyperbolic paraboloid, Z equals Y square

In this question, we're going to prove that the hyperboloid of one sheet is a, no, we're going t

Question

Start with the hyperbolic paraboloid surface z = f(x, y) = y^2 âˆ’ x^2, and an arbitrary point P(a, b, c) on the surface. A) What is c in terms of a and b? B) Find the scalar equation of the tangent plane at P, and write it as a function z = T(x, y). C) Equating f(x, y) = T(x, y) gives a quadratic expression for (x, y) values where the hyperbolic paraboloid surface intersects the tangent plane. Completing the squares in this expression leads to a difference of squares that nicely factors into the form P1(x, y)P2(x, y) = 0. D) Choose one of these factors to be zero, say P1(x, y) = 0. It is the equation of a plane. The line of intersection of this plane with the tangent plane z = T(x, y) gives one of these magical lines â€” construct the parametric equations for this line. (Hints: see example #7 in Section 12.5, and point P is on the line.) E) Verifying that every point on this line lies on the surface z = y^2 âˆ’ x^2 checks your math.

          Start with the hyperbolic paraboloid surface z = f(x, y) = y^2 âˆ’ x^2, and an arbitrary point P(a, b, c) on the surface.
A) What is c in terms of a and b?
B) Find the scalar equation of the tangent plane at P, and write it as a function z = T(x, y).
C) Equating f(x, y) = T(x, y) gives a quadratic expression for (x, y) values where the hyperbolic paraboloid surface intersects the tangent plane. Completing the squares in this expression leads to a difference of squares that nicely factors into the form P1(x, y)P2(x, y) = 0.
D) Choose one of these factors to be zero, say P1(x, y) = 0. It is the equation of a plane. The line of intersection of this plane with the tangent plane z = T(x, y) gives one of these magical lines â€” construct the parametric equations for this line. (Hints: see example #7 in Section 12.5, and point P is on the line.)
E) Verifying that every point on this line lies on the surface z = y^2 âˆ’ x^2 checks your math.

Added by Rebecca S.

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