Question #194836

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.


1
Expert's answer
2021-05-18T18:06:45-0400


i) TRUE


Explanation:


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



r=acosθ+bsinθr=a\cos \theta+b\sin \theta

Multiply both sides by r



r2=arcosθ+brsinθr^2=ar\cos\theta+br\sin\theta

Convert to cartesian coordinates



x2+y2=ax+byx^2+y^2=ax+by

Complete the square



x2ax+a24+y2by+b24=a24+b24x^2-ax+{a^2 \over 4}+y^2-by+{b^2 \over 4}={a^2 \over 4}+{b^2 \over 4}(xa2)2+(yb2)2=a24+b24(x-{a \over 2})^2+(y-{b \over 2})^2={a^2 \over 4}+{b^2 \over 4}

Therefore, the equation r = r = acos θ\theta + bsin θ\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 = cosα\cos\alpha = 1 \ ( √ 12 + 0.52 + 02) = 2 / √5

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

n = cosγ\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.






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