5. Consider the point P with coordinates (a, b, c) on the surface of the hyperbo

Question

5. Consider the point P with coordinates (a, b, c) on the surface of the hyperbolic paraboloid described by z-2-a2 (a) Determine z(t) such that the straight line r(t) = (a+1) (b+1) j + zit) k lies entirely on the surface and passes through the point P. Simplify your result so that only the terms a, b, and c appear. That is, terms such as a2, or a3, do not appear (b) A ruled quadric surface is a quadric surface that can be generated by the continuous motion of a straight line. Clearly, based on your results from part (a), any point on a hyperbolic paraboloid has at least one straight line that passes through it, but is the hyperbolic paraboloid a ruled quadric surface? If so, what motion of a straight line will generate the hyperbolic paraboloid? (c) Are there any other ruled quadric surfaces? If so, which surfaces are they, and how are they generated?

Explanation / Answer

z = y^2 - x^2

z = (b + t)^2 - (a + t)^2

z = b^2 + 2bt + t^2 - a^2 - 2at - t^2

z = b^2 - a^2 + 2bt - 2at

Now, we have a point (a,b,c)
So, c = b^2 - a^2 using z = y^2 - x^2

So, we have
z = c + 2bt - 2at -----> ANSWER

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A hyperbolic paraboloid is a DOUBLY RULED SURFACE

If we have a Straight Rod Passing Through Curved Hole,
then the resulting picture is a hyperbolic paraboloid

Here's the link :
https://www.youtube.com/watch?v=IXQ1ZWBhnBk

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c)

Of the non-degenerate quadratic surfaces, the elliptic (and usual) cylinder, hyperbolic cylinder, elliptic (and usual) cone are ruled surfaces, while the one-sheeted hyperboloid and hyperbolic paraboloid are doubly ruled surfaces.

Elliptic cylinder :
You just need to take a straight line and move it round and round in ellipses keeping the line vertical. The top point of the line will traverse an ellipse and the line itself will represent the heoght of the cylinder.

Cylinder :
You just need to take a straight line and move it round and round in circles keeping the line vertical. The top point of the line will traverse a circle and the line itself will represent the height of the cylinder.

Cone :
Take a slanted line.
Fix the top of the line at a point and move the bottom of the line round and round in circles and we get the cone.

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