Question

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.

Answer 1

i)

$$r=a(\dfrac{x}{r})+b(\dfrac{y}{r})$$

$$r^2=ax+by$$

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

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

This is the equation of circle.

Statement is True.

ii) Let the angles made by the line with positive x axis, y axis and z axis be $\alpha, \beta, \gamma,$ the direction cosines of the lines are $(\cos \alpha, \cos \beta, \cos \gamma)$

Given $\cos \alpha=1, \cos \beta= 1/2, \cos \gamma=0.$

Then $\alpha=0\degree , \beta=60\degree, \gamma=90\degree.$

Statement is False.

iii) Conic section can be parabola, hyperbola. ellipse (circle)

Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics.

Statement is True.

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

three mutually perpendicular tangent planes and vice versa.

Statement is True.

v) Transform the equations:

$$2x^2 + y^2 + 3z^2 + 4x + 4y + 18z + 34 = 0$$

$$2(x^2 +2x+1)-2+ (y^2+4y+4)-4$$

$$+ 3(z^2 +6z+9)-27 + 34 = 0$$

$$2(x+1)^2+ (y+2)^2+3(z+3)^2=-1$$

$$2x^ 2 −y^2 = 4y−4z−4x$$

$$4z=-2(x^2+2x+1)+2+(y^2+4y+4)-4$$

$$4z=-2(x+1)^2+(y-2)^2-2$$

$$z=-\dfrac{1}{2}(x+1)^2+\dfrac{1}{4}(y-2)^2-\dfrac{1}{2}$$

The first of these hasn’t any solution and the second is an equation of the saddle-shaped structure (known as a hyperbolic paraboloid or an anticlastic surface). Hence, the equations do not represent a real conic.

Statement is False.