The angle between the asymptotes of the hyperbola x² - 3y² = 3 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

If the area of the triangle formed by the pair of lines 8x² - 6xy + y² = 0 and the line 2x + 3y = a is 7, then a is equal to

• 14

• $14\sqrt{2}$

• $28\sqrt{2}$

• 28

If the line x + 3y = 0 is the tangent at (0, 0) to the circle of radius 1, then the centre of one such circle is

• (3, 0)

• $\left(\frac{- 1}{\sqrt{10}}, \frac{3}{\sqrt{10}}\right)$

• $\left(\frac{3}{\sqrt{10}}, \frac{- 3}{\sqrt{10}}\right)$

• $\left(\frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}}\right)$

A circle passes through the point (3, 4) and cuts the circle x² + y² = a² orthogonally; the locus of its centre is a straight line. If the distance of this straight line from the origin is 25, then a is equal to

• 250

• 225

• 100

• 25

355.

The equation to the line joining the centres of the circles belonging to the coaxial system of circles 4x² + 4y² - 12x + 6y - 3 + $\mathrm{\lambda }$(x + 2y - 6) = 0 is

• 8x - 4y - 15 = 0

• 8x - 4y + 15 = 0

• 3x - 4y - 5 = 0

• 3x - 4y + 5 = 0

Let x + y = k be a normal to the parabola y² = 12x. If p is length of the perpendicular from the focus of the parabola onto this normal, then 4k - 2p² is equal to

• 1

• 0

• - 1

• 2

If the line 2x + 5y = 12 intersects the ellipse 4x² + 5y² = 20 in two distinct points A and B,then mid-point of AB is

• (0, 1)

• (1, 2)

• (1, 0)

• (2, 1)

Equation of one of the tangents passing through(2, 8) to the hyperbola 5x² - y² = 5 is

• 3x + y - 14 = 0

• 3x - y + 2 = 0

• x + y + 3 = 0

• x - y + 6 = 0

The area (in sq units) of the equilateral triangle formed by the tangent at ($\sqrt{3}$, 0) to the hyperbola x² - 3y² = 3 with the pair of asymptotes of the hyperbola is

• $\sqrt{2}$

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• $2\sqrt{3}$

The radius of the circle r = $12\cos \left(\mathrm{\theta }\right) + 5\sin \left(\mathrm{\theta }\right) \mathrm{IS}$

• $\frac{5}{12}$

• $\frac{17}{2}$

• $\frac{15}{2}$

• None of these