For different values of a, the locus of the point of intersection of the two straight lines $\sqrt{3}\mathrm{x} - \mathrm{y} - 4\sqrt{3}\mathrm{\alpha } = 0$ and $\sqrt{3}\mathrm{\alpha x} + \mathrm{\alpha y} - 4\sqrt{3} = 0$

• a hyperbola with eccentricity 2

• an ellipse with eccentricity $\sqrt{\frac{2}{3}}$

• a hyperbola with eccentricity $\sqrt{\frac{19}{16}}$

• an ellipse with eccentricity $\frac{3}{4}$

The circles x² + y² - 10x + 16 = 0 and x² + y² = a² intersect at two distinct points, if

• a < 2

• 2 < a < 8

• a > 8

• a = 2

For the two circles x² + y² = 16 and x² + y² - 2y = 0 there is/are

• one pair of common tangents

• only one common tangent

• three common tangents

• no common tangent

The equation of the tangent to the conic x² - y² - 8x + 2y + 11 = 0 at (2, 1) is

• x + 2 = 0

• 2x + 1 = 0

• x + y + 1 = 0

• x - 2 = 0

The total number of tangents through the point ( 3, 5) that can be drawn to the ellipses 3x² + 5y² = 32 and 25x + 9y² = 450 is

• 0

• 2

• 3

• 4

The equation of chord of the circle x² + y² - 4x = 0, whose mid-point is (1, 0) is

• y = 2

• y = 1

• x = 2

• x = 1

The line y = 2t² intersects the ellipse $\frac{{\mathrm{x}}^2}{9} + \frac{{\mathrm{y}}^2}{4} = 1$ in real points, if

• $\left|\mathrm{t}\right| \le 1$

• $\left|\mathrm{t}\right| < 1$

• $\left|\mathrm{t}\right| > 1$

• $\left|\mathrm{t}\right| \ge 1$

The coordinates of the focus of the parabola described parametrically by x = 5t² + 2, y = 10t + 4 are

• (7, 4)

• (3, 4)

• (3, - 4)

• (- 7, 4)

The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is 90°. The eccentricity of the ellipse is

• 1/8

• 1/√3

• √(2/3)

• √(1/2)

120.

If the rate of increase of the radius of a circle is 5 cm/ s, then the rate of increase of its area, when the radius is 20 cm, will be

• 10$\mathrm{\pi }$

• 20$\mathrm{\pi }$

• 200$\mathrm{\pi }$

• 400$\mathrm{\pi }$