A circle S cuts three circles

$$\begin{gathered} {\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 2\mathrm{y} +\hspace{0.17em}4 = 0 \\ {\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{x} - 4\mathrm{y} + 1 = 0 \\ \mathrm{and} {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 +\hspace{0.17em}4\mathrm{x} +\hspace{0.17em}2\mathrm{y} +\hspace{0.17em}1 = 0 \end{gathered}$$

Orthogonally. Then the radius of S is

• $\sqrt{\frac{29}{8}}$

• $\sqrt{\frac{28}{11}}$

• $\sqrt{\frac{29}{7}}$

• $\sqrt{\frac{29}{5}}$

The distance between the vertex and the focus of the parabola x² - 2x + 3y - 2 = 0 is

• $\frac{4}{5}$

• $\frac{3}{4}$

• $\frac{1}{2}$

• $\frac{5}{6}$

If (x_1, y₁) and (x₂, y₂) are the end points of a focal chord of the parabola y² = 5x, then 4x₁x₂ + y₁y₂, is equal to

• 25

• 5

• 0

• $\frac{5}{4}$

The distance between the focii of the ellipse
x = 3cos$\left(\mathrm{\theta }\right)$, y = 4sin$\left(\mathrm{\theta }\right)$ is

• $2\sqrt{7}$

• $7\sqrt{2}$

• $\sqrt{7}$

• $3\sqrt{7}$

The equations of the latus rectum of the ellipse
9x² + 25y² - 36x + 50y - 164 = 0 are

• x - 4 = 0, x + 2 = 0

• x - 6 = 0, x + 2 = 0

• x + 6 = 0, x - 2 = 0

• x + 4 = 0, x + 5 = 0

The values of m for which the line y = mx + 2
becomes a tangent to the hyperbola 4x² - 9y² = 36 is

• $\pm \frac{2}{3}$

• $\pm \frac{2\sqrt{2}}{3}$

• $\pm \frac{8}{9}$

• $\pm \frac{4\sqrt{2}}{3}$

The equation of the common tangent drawn to the curves y = 8x and xy = - 1 is

• y = 2x + 1

• 2y = x + 6

• y = x + 2

• 3y = 8x + 2

The area included between the parabola $\mathrm{y} = \frac{{\mathrm{x}}^2}{4\mathrm{a}} \mathrm{and} \mathrm{the} \mathrm{curve} \mathrm{y} = \frac{8{\mathrm{a}}^3}{\left({\mathrm{x}}^2 + 4{\mathrm{a}}^2\right)} \mathrm{is}$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } + \frac{2}{3}\right)$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } - \frac{8}{3}\right)$

• ${\mathrm{a}}^2\left(\mathrm{\pi } + \frac{4}{3}\right)$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } - \frac{4}{3}\right)$

If a circle with radius 2.5 units passes through the points (2, 3) and (5, 7), then its centre is

• (1 5, 2)

• (7, 10)

• (3, 4)

• (3 5, 5)

The circumcentre of the triangle formed by the points (1, 2, 3) (3, - 1, 5), (4, 0, - 3) is

• (1, 1, 1)

• (2, 2, 2)

• (3, 3, 3)

• $\left(\frac{7}{2}, \frac{ - 1}{2}, 1\right)$