$$\begin{gathered} \mathrm{If} {\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 2\mathrm{y} +\hspace{0.17em}5 = 0 \mathrm{and} {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 - 6\mathrm{x} - 4\mathrm{y} - 3 = 0 \mathrm{are} \mathrm{members} \mathrm{of} \mathrm{a} \mathrm{coaxial} \mathrm{system} \mathrm{of} \mathrm{circles}, \\ \mathrm{then} \mathrm{the} \mathrm{centre} \mathrm{of} \mathrm{a} \mathrm{circle} \mathrm{in} \mathrm{the} \mathrm{system} \mathrm{is} \end{gathered}$$

• ( - 5, - 6)

• (5, 6)

• (3, 5)

• ( - 8, - 13)

Equation of the locus of the centroid of the triangle whose vertices are (acos(k), asin(k)), [bsin(k), - bcos(k)) and (1, 0), where k is a parameter, is

• ${\left(1 - 3\mathrm{x}\right)}^2 + 9{\mathrm{y}}^2 = {\mathrm{a}}^2 + {\mathrm{b}}^2$

• ${\left(3\mathrm{x} - 1\right)}^2 + 9{\mathrm{y}}^2 = 2{\mathrm{a}}^2 + 2{\mathrm{b}}^2$

• ${\left(3\mathrm{x} + 1\right)}^2 + {\left(3\mathrm{y}\right)}^2 = 3{\mathrm{a}}^2 + 3{\mathrm{b}}^2$

• ${\left(3\mathrm{x} +\hspace{0.17em}1\right)}^2 + {\left(3\mathrm{y}\right)}^2 = 3{\mathrm{a}}^2 + 3{\mathrm{b}}^2$

A circle S = 0 with radius $\sqrt{2}$ touches the line x + y - z = 0 at(1, 1). Then, the length of the tangent drawn from the point(1, 2) to S = 0 is

• 1

• 2

• $\sqrt{3}$

• 2

The normal drawn at P(- 1, 2) on the circle x² + y² - 2x - 2y - 3 = 0 meets the circle at another point Q. Then the coordinates of Q are

• (3, 0)

• ( - 3, 0)

• (2, 0)

• ( - 2, 0)

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

• 0

• 1

• 2

• 3

The angle between the, tangents drawn from the origin to the circle x² + y² + 4x - 6y + 4 = 0 is

• ${\tan }^{-1}\left(\frac{5}{13}\right)$

• ${\tan }^{-1}\left(\frac{5}{12}\right)$

• ${\tan }^{-1}\left(\frac{- 12}{5}\right)$

• ${\tan }^{-1}\left(\frac{13}{5}\right)$

If the angle between the circles x² + y² - 2x - 4y + c = 0 and x² + y² - 4x - 2y + 4 = 0 is 60°, then c is equal to

• $\frac{3 \pm \sqrt{5}}{2}$

• $\frac{6 \pm \sqrt{5}}{2}$

• $\frac{9 \pm \sqrt{5}}{2}$

• $\frac{7 \pm \sqrt{5}}{2}$

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

• $\frac{5}{4}$