Answer to Question #91820 in Analytic Geometry for Ra

Question

Which of the following statements are true and which are false? Please give reasons for the answers.
1.) 4x²-9y²+z²+36=0 represents a hyperboloid of one sheet.
2.) The intersection of any plane with an ellipsoid is an ellipse.
3.) No plane passes through the points (1,2,3), (1,-1,0) and (1,1,2).
4.) The circle with centre (a,0) and radius a, where a>0, touches all the sides of the square x = 0,x =a,y=±a.
5.) If the projection of a line segment AB on a line L is 0, then AB lies in L.

Answer 1

1) The statement is false:

$4x²-9y²+z²+36=0$

$4x²-9y²+z²=-36$

$x²/9-y²/4+z²/36=-1$

$\frac{x²}{3^2}+\frac{z²}{6^2}-\frac{y²}{2^2}=-1.$

This is an equation of the two-sheet hyperboloid (−1 in the right-hand side of the equation).

2) The statement is true.

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of an affine transformation. Since the intersection of a sphere by a plane is a circle, and the affine transformation of the circle is an ellipse, so the intersection of any plane with an ellipsoid is an ellipse.

3) The statement is false because at least one plane can be drawn through any three points.

4) The statement is false. The line x=a passes through the center of the circle.

5) The statement is false. If the projection of a line segment AB on a line L is 0, then AB ⊥ L.