Mathematics : Three Dimensional Geometry

The equation of the plane containing the line $\frac{\mathrm{x} + 1}{- 3} = \frac{\mathrm{y} - 3}{2} = \frac{\mathrm{z} + 2}{1}$ and the point (0, 7, - 7) is

• x + y + z = 1

• x + y + z = 2

• x + y + z = 0

• None of these

Angle of intersection of the curve r = $\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right)$ and r = 2$\sin \left(\mathrm{\theta }\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• None of these

A point on XOZ - plane divides the join of (5, - 3, - 2) and (1, 2, - 2) at

• $\left(\frac{13}{5}, 0, - 2\right)$

• $\left(\frac{13}{5}, 0, 2\right)$

• (5, 0, 2)

• (5, 0, - 2)

If the line $\overrightarrow{\mathrm{OR}}$ makes angles ${\mathrm{\theta }}_1, {\mathrm{\theta }}_2, {\mathrm{\theta }}_3$, with the planes XOY, YOZ, ZOX respectively, then ${\cos }^2\left({\mathrm{\theta }}_1\right) + {\cos }^2\left({\mathrm{\theta }}_2\right) + {\cos }^2\left({\mathrm{\theta }}_3\right)$, is equal to

• 1

• 2

• 3

• 4

Joint equation of pair of lines through (3, - 2) and parallel to x² - 4xy + 3y² = 0 is

• x² + 3y² - 4xy - 14x + 24y + 45 = 0

• x² + 3y² + 4xy - 14x + 24y + 45 = 0

• x² + 3y² + 4xy - 14x + 24y - 45 = 0

• x² + 3y² + 4xy - 14x - 24y - 45 = 0

Equation of the plane passing through (- 2, 2, 2) and (2, - 2, - 2) and perpendicular to the plane 9x - 13y - 3z = 0 is

• 5x + 3y + 2z = 0

• 5x - 3y + 2z = 0

• 5x - 3y - 2z = 0

• 5x + 3y - 2z = 0

If 'f' is the angle between the lines ax² + 2hxy + by² = 0, then angle between x² + 2xy sec$\left(\mathrm{\theta }\right)$ + y² = 0 is

• $\mathrm{\theta }$

• $2\mathrm{\theta }$

• $\frac{\mathrm{\theta }}{2}$

• $3\mathrm{\theta }$

The equation of the plane which passes through (2, - 3, 1) and is normal to the line joining the points (3, 4, - 1) and (2, - 1, 5), is given by

• x + 5y - 6z + 19 = 0

• x - 5y + 6z - 19 = 0

• x + 5y + 6z + 19 = 0

• x - 5y - 6z - 19 = 0

The angle between the lines x² - xy - 6y² - 7x + 31y - 18 = 0 is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

The equation of the lines passing through the origin and having slopes 3 and - $\frac{1}{3}$, is

• 3y² + 8xy - 3x² = 0

• 3x² + 8xy + 3y² = 0

• 3y² - 8xy - 3x² = 0

• 3x² + 8xy - 3y² = 0