If $\mathrm{\theta }$ is the angle between the tangents from(- 1, 0) to the circle x² + y² - 5x + 4y - 2 = 0, then $\mathrm{\theta }$ is equal to

• $2{\tan }^{-1}\left(\frac{7}{4}\right)$

• ${\tan }^{-1}\left(\frac{7}{4}\right)$

• $2{\cos }^{-1}\left(\frac{7}{4}\right)$

• ${\cot }^{-1}\left(\frac{7}{4}\right)$

If 2x + 3y + 12 = 0 and x - y + 4$\mathrm{\lambda }$ = 0 are conjugate with respect to the parabola y = 8x, then $\mathrm{\lambda }$ is equal to

• 2

• - 2

• 3

• - 3

For an ellipse with eccentricity $\frac{1}{2}$ the centre is at the origin. If one directrix is x = 4, then the equation of the ellipse is

• $3{\mathrm{x}}^2 + 4{\mathrm{y}}^2 = 1$

• $3{\mathrm{x}}^2 + 4{\mathrm{y}}^2 = 12$

• $4{\mathrm{x}}^2 + 3{\mathrm{y}}^2 = 1$

• $4{\mathrm{x}}^2 + 3{\mathrm{y}}^2 = 12$

The distance between the foci of the hyperbola x² - 3y² - 4x - 6y - 11 = 0 is

• 4

• 6

• 8

• 10

The radius of the circle with the polar equation

r² - 8r($\sqrt{3}$cos$\mathrm{\theta }$ + sin$\mathrm{\theta }$) + 15 = 0 is

• 8

• 7

• 6

• 5

The area (in square unit) of the circle which touches the lines 4x + 3y = 15 and 4x + 3y = 5 is

• 4$\mathrm{\pi }$

• $\mathrm{3\pi }$

• $\mathrm{2\pi }$

• $\mathrm{\pi }$

The equations of the circle which pass through the origin and makes intercepts of length 4 and 8 on the x and y-axes respectively are

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 4\mathrm{x} \pm 8\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 2\mathrm{x} \pm 4\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 8\mathrm{x} \pm 16\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm \mathrm{x} \pm \mathrm{y} = 0$

The locus of centre of a circle which passes through the origin and cuts off a length of 4 unit from the line x = 3 is

• y² + 6x = 0

• y² + 6x = 13

• y² + 6x = 10

• x² + 6y = 13

The diameters of a circle are along 2x +y - 7 and x + 3y - 11 = 0. Then, the equation of this circle, which also passes through (5, 7) is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 6\mathrm{y} - 16 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 6\mathrm{y} - 20 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 6\mathrm{y} - 12 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 4\mathrm{x} + 6\mathrm{y} - 12 = 0$

$$\begin{gathered} \mathrm{The} \mathrm{point} \left(3, - 4\right) \mathrm{lies} \mathrm{on} \mathrm{both} \mathrm{the} \mathrm{circles} \\ {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 - 2\mathrm{x} + 8\mathrm{y} +\hspace{0.17em}13 = 0 \mathrm{and} {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 -\hspace{0.17em}4\mathrm{x} +\hspace{0.17em}6\mathrm{y} + 11 = 0, \\ \mathrm{Then}, \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{circles} \mathrm{is} \end{gathered}$$

• $60°$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$

• ${\tan }^{-1}\left(\frac{3}{5}\right)$

• $135°$