Answer to Question #194836 in Analytic Geometry for Shubham

Question

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

answer. (20)

i) The equation r = acosθ +bsinθ 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.

Answer 1

i) TRUE

Explanation:

The equation r = acos $\theta$ + bsin $\theta$ represents a circle. 

$$r=a\cos \theta+b\sin \theta$$

Multiply both sides by r

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

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 = r = acos $\theta$ + bsin $\theta$ represents a circle. 

ii) FALSE

Explanation:

 If 1,1/2,0 are direction ratios of a line, then the direction cosines will be as follows

l = $\cos\alpha$ = 1 \ ( √ 1² + 0.5² + 0²) = 2 / √5

m = $\cos\beta$ = 0.5 \ ( √ 1² + 0.5² + 0²) = 1 / √5

n = $\cos\gamma$ = 0 \ ( √ 1² + 0.5² + 0²) = 0

So the line makes 26.77$\degree$ with the x - axis , 63.48$\degree$ with the y - axis and 90$\degree$ with the z - axis.

iii) TRUE

Explanation:

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).

iv) TRUE

EXPLANTION:

If a cone has three mutually perpendicular generators then its reciprocal cone has

three mutually perpendicular tangent planes and vice versa.