If the point of intersection of the tangents drawn at the points where the line 5x + y + 1 = 0 cuts the circle x² + y² - zx - 6y - 8 = 0 is (a, b), then 5a + b =

• 3

•  - 44

•  - 1

• 4

If 2kx + 3y - 1 = 0, 2x + y + 5 = 0 are conjugate lines with respect to the circle x² + y² - 2x - 4y - 4 = 0, then k =

• 3

• 4

• 1

• 2

The equations of the parabola whose axis is parallel to the X-axis and which passes through the points (- 2, 1), (1, 2)(- 1, 3) is

• $18{\mathrm{y}}^2 - 12\mathrm{x} - 21\mathrm{y} - 21 = 0$

• $5{\mathrm{y}}^2 + 2\mathrm{x} - 21\mathrm{y} + 20 = 0$

• $15{\mathrm{y}}^2 + 12\mathrm{x} - 11\mathrm{y} + 20 = 0$

• $25{\mathrm{y}}^2 - 2\mathrm{x} - 65\mathrm{y} + 36 = 0$

The angle between the two circles, each passing through the centre of the other is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• $\mathrm{\pi }$

$\mathrm{If} {\log }_{\frac{1}{\sqrt{3}}}\left\{\frac{{\left|\mathrm{z}\right|}^2 - \left|\mathrm{z}\right| + 1}{2 + \left|\mathrm{z}\right|}\right\} > - 2, \mathrm{then} \mathrm{z} \mathrm{lies} \mathrm{inside}$

• a triangle

• an ellipse

• a circle

• a square

416.

A circle having centre at the origin passes through the three vertices of an equilateral triangle the length of its median being 9 units. Then the equation of that circle is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 9$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 18$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 36$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 81$

A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola $\frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{2} = 1$  at the point (x₁, y₁). Then ${{\mathrm{x}}_1}^2 + 5{{\mathrm{y}}_1}^2$ is equal to :

• 10

• 5

• 8

• 6

The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² + y² – 2x – 4y + 4 = 0 at two distinct points is .... 

For some $\mathrm{\theta } \in \left(0, \frac{\mathrm{\pi }}{2}\right)$, if the eccentricity of the hyperbola, ${\mathrm{x}}^2 - {\mathrm{y}}^2{\sec }^2\left(\mathrm{\theta }\right) = 10$ is 5 times the eccentricity of the ellipse , ${\mathrm{x}}^2{\sec }^2\left(\mathrm{\theta }\right) + {\mathrm{y}}^2 = 5$, then the length of the latus rectum of the ellipse, is

• $\frac{2\sqrt{5}}{3}$

• $2\sqrt{6}$

• $\frac{4\sqrt{5}}{3}$

• $\sqrt{30}$

The area (in sq. units) of an equilateral triangle inscribed in the parabola y² = 8x, with one of its vertices on the vertex of this parabola is

• $64\sqrt{3}$

• $192\sqrt{3}$

• $128\sqrt{3}$

• $256\sqrt{3}$