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351.
The angle between the asymptotes of the hyperbola x² - 3y² = 3 is

$\frac{π}{6}$

$\frac{π}{4}$

$\frac{π}{3}$

$\frac{π}{2}$

$\frac{π}{3}$

$Given equation of hyperbola is x^{2} - 3 y^{2} = 1 \Rightarrow \frac{x^{2}}{3} - \frac{y^{2}}{1} = 1 Here, a^{2} = 3 and b^{2} = 1, a > b Now, equation of asymptote of this hyperbola is, y = \pm \frac{b}{a} x \Rightarrow y = \pm \frac{1}{\sqrt{3}} x \Rightarrow y = \frac{x}{\sqrt{3}} . . . (i) and y = \frac{- x}{\sqrt{3}} . . . (ii) Let slope of asymptote (i) is m_{1}, = \frac{1}{\sqrt{3}} and slope of asymptote (ii) is m_{2} = \frac{- 1}{\sqrt{3}} Let θ be the angle between both asymptote, then \tan (θ) = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}| = |\frac{\frac{1}{\sqrt{3} + \frac{1}{\sqrt{3}}}}{1 - \frac{1}{3}}| \Rightarrow \tan (θ) = |\frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}}| = \sqrt{3} \Rightarrow \tan (θ) = \tan (60 °) \Rightarrow θ = \frac{π}{3}$

352.

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

353.

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

354.

A circle passes through the point (3, 4) and cuts the circle x²+ y²= a²orthogonally; 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

355.

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

8x - 4y - 15 = 0
8x - 4y + 15 = 0
3x - 4y - 5 = 0
3x - 4y + 5 = 0

356.

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

357.

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)

358.

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

359.

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