The value of k so that x² + y² + kx + 4y + 2 = 0 and 2(x² + y) - 4x - 3y + k = 0 cut orthogonally is

• $\frac{10}{3}$

• $\frac{- 8}{3}$

• $\frac{- 10}{3}$

• $\frac{8}{3}$

If the lines $\frac{\mathrm{x} - 1}{2} = \frac{\mathrm{y} + 2}{3} = \frac{\mathrm{z} - 1}{4}$ and $\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - \mathrm{k}}{2} = \frac{\mathrm{z}}{1}$ intersect, then the value of k is

• 3/2

• 7/2

• - 2/7

• - 3/2

If $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \mathrm{a\delta } \mathrm{and} \mathrm{\beta } + \mathrm{\gamma } + \mathrm{\delta } = \mathrm{b\alpha }, \mathrm{\alpha } \mathrm{and} \mathrm{\delta }$ are non-collinear, then $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } + \mathrm{\delta }$ equals

• 0

• $\mathrm{a\alpha }$

• b$\mathrm{\delta }$

• (a + b)$\mathrm{\gamma }$

The two lines x = my + n, z = py + q and x = m'y + n', z =p'y + q' are perpendicular to each other, if

• mm' + pp' = 1

• $\frac{\mathrm{m}}{\mathrm{m}\text{'}} + \frac{\mathrm{p}}{\mathrm{p}\text{'}} = - 1$

• $\frac{\mathrm{m}}{\mathrm{m}\text{'}} + \frac{\mathrm{p}}{\mathrm{p}\text{'}} = 1$

• mm' + pp' = - 1

The shortest distance between the lines $\frac{\mathrm{x} - 7}{3} = \frac{\mathrm{y} + 4}{- 16} = \frac{\mathrm{z} - 6}{7}$ and $\frac{\mathrm{x} - 10}{3} = \frac{\mathrm{y} - 30}{8} = \frac{4 - \mathrm{z}}{5}$ is

• $\frac{234}{7}$ units

• $\frac{288}{21}$ units

• $\frac{221}{3}$ units

• $\frac{234}{21}$ units

Two lines $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - \mathrm{k}}{2} = \mathrm{z}$ intersect at a point, if k is equal to

• $\frac{2}{9}$

• $\frac{1}{2}$

• $\frac{9}{2}$

• $\frac{1}{6}$

The angle between lines joining the origin to the point of intersection of the line $\sqrt{3}$x + y = 2 and the curve y² - x² = 4 is

• ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

The line joining two points A(2, 0), B(3, 1) is rotated about A in anti-clockwise direction through an angle of 15°. The equation of the line in the now position, is

• √3x - y - 2√3 = 0

• x - 3√y - 2 = 0

• √3x + y - 2√3 = 0

• x - √3y - 2 = 0

The line 2x + $\sqrt{6}$y = 2 is a tanent to the curve x² - 2y² = 4. The point of contact is

• $\left(4, - \sqrt{6}\right)$

• $\left(7, - 2\sqrt{6}\right)$

• (2, 3)

• $\left(\sqrt{6}, 1\right)$

A straight line through the point A (3, 4) is such that its intercept between the axes is bisected at A. Its equation is

• 3x - 4y + 7 = 0

• 4x + 3y = 24

• 3x + 4y = 25

• x + y = 7