Analytic Geometry Answers

2x + 3y = 7z represents a line in three-dimensional space.
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

1/√2, 1/√3, 1/√5 form the direction cosines of a line.
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

Under a rotation of axes, a parabola can become a hyperbola.
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

Center of the circle (0,0) is tangent to the line x-3y =6

Find the equation of a locus of a point so that the square of its distance from (3,-3) is always numerically equal to the slope of the line joining it to the same point

Obtain the equation of the sphere having center on the line x/3= y/2= z/-5 and passing through the points
�(0,-2,-4) and (2,-1,-1)

Find the coordinates of the foot of the perpendicular from (-2,6) on the line
2x+3y-1=0. �

Obtain the equation of the line passing through (1,-1,2) having direction ratios (2,0,1)

A ̅=5i+7j+8k

B ̅= 5i+2j-5k


Use your understanding of vector analysis to complete the following:

Calculate a resultant vector which would theoretically represent a single force that could replace the two force vectors A and B while giving the same support to the structure.

Calculate the modulus of all three forces.
Determine the value of the dot-product (scalar product) of vectors A and B
Calculate the angle between the vectors A and B
Determine the directional cosine angles of the resultant vector with respect to the x, y and z axes.

Find the equation of circle with center at the origin and tangent to the line 2x - 5y =8.