Analytic Geometry Answers

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.

Which of the following statements are true and which are false? Please give reasons for the answers.
1.) The equation r=acos(θ+α)+bsin(θ+α) represents a circle.
2.) The direction ratios of the line x-5=5-y, z=5 are 1,1,5.
3.) The intersection of a plane and a cone can be a pair of lines.
4.) The angle between the planes x+2y+2z=5 and 2x+2y+3=0 is 60º
5.) The equations 2x²+y²+3z²+4x+4y+18z+34=0, 2x²-y²=4y-4y-4x represent a real conic.

If the normal at one extremity of latus rectum of an ellipse passes through one extremity of the minor axis, prove that the eccentricity e is given by e⁴+e²-1=0

Let S≡4x²−9y²−36=0 and S'≡y²−4x=0 be two conics. Under what conditions on k, will the conic S+kS'= 0 represent: i) an ellipse? ii) a hyperbola?

How to prove that the path traced by the foot of the perpendicular from the focus of a parabola on any tangent to the parabola is the tangent at its vertex.

To find the equation of the right circular cone whose vertex is (1,-1,2), the axis is
x-1/2 = y+1/1 = z-2/-2
and the semi-vertical angle is 45º

The sketch shows the hyperbola defined by y the straight line defined by y ; a circle with centre at P, touching the r-axis and y-axis at R and S, respectively; and the straight line through the points T, S and R. The line joining T and Q is parallel to the y-axis. 2.1 Determine the coordinates of P and Q (5) 2.2 Determine the coordinates of S and R (2) 2.3 Find the radius of the circle, and write down the equation of the circle (3) 2.4 Determine the equation of the line through T, S and R. (5) 2.5 Calculate the length of the line TQ (5)

Prove that the planes
7x+4y-4z+30=0, 36x-51y+12z+17=0,14x+8y-8z-12=0 and 12x-17y+4z-3=0 form the four faces of a cuboid.

Show that if ux+vy+wz=p is a tangent plane to the paraboloid ax²+by²=2z, then u²/a+v²/b+2pw=0

Show that the angle between the two lines in which the plane x-y+2z=0 intersects the cone x²+y²-4z²+6yz=0 is tan⁻¹(√(6/7))