Mathematics : Straight Lines

The angle between the lines 2x = 3y = - z and 6x = - y = - 4z is

• 0°

• 30°

• 45°

• 90°

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the angles which a straight line makes with the positive drrection of the axes, then ${\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• 1

• 0

• 5

• 2

Equation of the pair of straight lines bisecting the angles between the lines represented by ax² + hxy + by² = 0

• $\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{\mathrm{a} - \mathrm{b}} = \frac{2\mathrm{xy}}{\mathrm{h}}$

• $\frac{{\mathrm{x}}^2 + {\mathrm{y}}^2}{\mathrm{a} + \mathrm{b}} = \frac{2\mathrm{xy}}{\mathrm{h}}$

• $\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{\mathrm{a} - \mathrm{b}} = \frac{\mathrm{xy}}{\mathrm{h}}$

• None of these

Angle between any two diagonals ofa cube is

• $\frac{\mathrm{\pi }}{3}$

• ${\cos }^{-1}\left(\frac{1}{3}\right)$

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

• None of the above

The equation of lines joining the origin to the points of intersection of y = x + 3 and 4x² + 4y² = 1 is

• 36(x² + y²) = (x - y)²

• 12(x² + y²) = (x + y)²

• 9(x² + y²) = 4(x - y)²

• None of the above

The distance between the lines 3x + 4y = 9 and 6x + 8y = 15 will be

• $\frac{3}{2}$

• $\frac{3}{8}$

• $\frac{3}{10}$

• 6

If the algebraic sum of the perpendicular distances from the points (2, 0), (0, 2) and (1, 1) on a variable line is zero, then the line will pass through the fixed point

• (1, 2)

• (1, 1)

• (0, 0)

• (2, 1)

The point on x² = 2y, which is closest to the point (0, 5), will be

• $\left(2\sqrt{2}, 0\right)$

• (0, 0)

• (2, 2)

• None of these

If a + b + c = 0, the straight line 2ax + 3by + 4c = 0 passes through the fixed point

• $\left(2, \frac{4}{3}\right)$

• (2, 2)

• $\left(\frac{4}{3}, \frac{4}{3}\right)$

• No such fixed point

The orthocentre of the triangle formed by tie lines xy = 0 and x + y = 1 is

• $\left(\frac{1}{2}, \frac{1}{2}\right)$

• $\left(\frac{1}{3}, \frac{1}{3}\right)$

• (0, 0)

• $\left(\frac{1}{4}, \frac{1}{4}\right)$