Question

Which of the following statements are true and which are false? Give reasons for your
answer.
i) The equation r = acos(theta+alpha)+bsin(theta +alpha) represents a circle.
ii) The direction ratios of the line x-5 = 5-y; z = 5 are 1;1;5:
iii) The intersection of a plane and a cone can be a pair of lines.

Answer 1

i) The equation r = acos(theta+alpha)+bsin(theta +alpha) represents a circle. 

$$r=a\cos(\theta+\alpha)+b\sin(\theta+\alpha)$$

Multiply both sides by r

$$r^2=ar\cos(\theta+\alpha)+br\sin(\theta+\alpha)$$

Convert to cartesian coordinates

$$x^2+y^2=ax+by$$

Complete the square

$$x^2-ax+{a^2 \over 4}+y^2-by+{b^2 \over 4}={a^2 \over 4}+{b^2 \over 4}$$

$$(x-{a \over 2})^2+(y-{b \over 2})^2={a^2 \over 4}+{b^2 \over 4}$$

Therefore, the equation r = acos(theta+alpha)+bsin(theta +alpha) represents a circle. 

True.

ii) The direction ratios of the line x-5 = 5-y; z = 5 are 1;1;5: 

The equation of the line

$${x-5 \over 1}={y-5 \over -1}={z-5 \over 0}$$

The direction ratios of the line are 1; -1; 0.

False.

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

If you intersect a cone with a plane, the intersection will be one of the following: a parabola, a circle, an ellipse, a hyperbola, a pair of lines (the plane must lie along the axis of the cone), a single line (plane is tangent to cone), or a single unique point (the plane must be perpendicular to the axis, passing through the center).

True.