Analytic Geometry Answers

Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

i) x² +y² +z² + x + y + z = 1

ii) 2x² +4xy+ xz−x−3y + 5z+ 3 = 0

iii) x² −y² −z² +xy + 4yz + x = 0

Prove that the plane 2x−3y+6z = 6 touches the conicoid 4x² −9y² +36z² = 36. Find the point of contact.

b) Identify and trace the conic x² −2xy+ y² −3x+ 2y+ 3 = 0.

a) Find the equation of the normal to the parabola y ² +4x = 0 at the point where the line y = x+c touches it.

Which of the following statements are true and which are false? Give reasons for your answer.

i) 4x ² −9y ² +z ² +36 = 0 represents a hyperboloid of one sheet

ii) The intersection of any plane with an ellipsoid is an ellipse

iii) No plane passes through the points (1,2,3),(1,−1,0) and (1,1,2).

iv) 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

v) 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? Give reasons for your answer.

i) The equation r = acos θ +b sin θ represents a circle.

ii) If 1,1/2,0 are direction ratios of a line, then the line makes an angle of 90◦ with the x-axis, an angle of 60◦ with the y-axis, and is parallel to the z-axis.

iii) The intersection of a plane and a cone can be a pair of lines. 

iv) If a cone has three mutually perpendicular generators then its reciprocal cone has three mutually perpendicular tangent planes. 

v) The equations 2x ² +y ² +3z ² +4x+4y+18z+34 = 0,2x ² −y ² = 4y−4y−4x represent a real conic.

Let a=(1,2,3), b=(-5,3,-2), c=(2,-4,1) be three points in R³. Find |2b - a + 3c|.

Explain Quadratic surfaces with diagram and at least one example in each case.

Find the vector equation of the plane determined by the points (-1,-2,1),(1,0,1) and (1,-1,1). Also check whether (1/2,1/2,1/2) lies on it.

Use the fact that

 x y 1

a1 b1 1 =0

a2 b2 1

to determine the equation of the line passing through the distinct points (a1, b1) and (a2, b2), where |·| stands for det(·), the determinant.