Introduction To Three Dimensional Geometry

The radius of the circle given by

x² + y² + z² + 2x - 2y - 2z - 19 = 0

• 4

• 3

• 2

• 1

If x-coordinate of a point P on the line joining the points Q(2, 2, 1) and R(5, 1, - 2) is 4, then the z-coordinate of P is

• - 2

• - 1

• 1

• 2

The equation of the sphere through the points (1, 0, 0), (0, 1, 0) and (1, 1, 1)and having the smallest radius

• $3\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2\right) - 4\mathrm{x} - 4\mathrm{y} - 2\mathrm{z} + 1 = 0$

• $2\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2\right) - 3\mathrm{x} - 3\mathrm{y} - \mathrm{z} + 1 = 0$

• $\left({\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2\right) - \mathrm{x} - \mathrm{y} - \mathrm{z} + 1 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 - 2\mathrm{x} - 2\mathrm{y} + 4\mathrm{z} + 1 = 0$

The perpendicular distance from the point $\left(1, \mathrm{\pi }\right)$ to the line joining (1, 0⁰) and $\left(1, \frac{\mathrm{\pi }}{2}\right)$ (in polar coordinates) is

• 2

• $\sqrt{3}$

• 1

• $\sqrt{2}$

$\mathrm{If} \frac{2{\mathrm{x}}^3 + {\mathrm{x}}^2 - 5}{{\mathrm{x}}^4 - 25} = \frac{\mathrm{Ax} + \mathrm{B}}{{\mathrm{x}}^2 - 5} + \frac{\mathrm{Cx} \hspace{0.17em}+ 1}{{\mathrm{x}}^2 +\hspace{0.17em}5}, \mathrm{then} \left(\mathrm{A}, \mathrm{B}, \mathrm{C}\right) = ?$

• (1, 1, 1)

• (1, 1, 0)

• (1, 0, 1)

• (1, 2, 1)

If in a $∆\mathrm{ABC}$, r₁ = 2r₂ = 3r₃, then the perimeter of the triangle is equal to

• 3a

• 3b

• 3c

• 3(a + b + c)

If the equation to the locus of points equidistant from the points (- 2, 3), (6, - 5) is ax + by + c = 0, where a > 0, then the ascending order of a, b, c is

• a, b, c

• c, b, a

• b, c, a

• a, c, b

If the point P on the curve, 4x² + 5y² = 20 is farthest from the point Q(0, – 4), then PQ² is equal to :

• 29

• 48

• 21

• 36

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that MD² + MC² is minimums is ___