Mathematics : Conic Section

The point (3, 4) is the focus and 2x - 3y + 5 = 0 is the directrix of a parabola. Its latus rectum is

• $\frac{2}{\sqrt{13}}$

• $\frac{4}{\sqrt{13}}$

• $\frac{1}{\sqrt{13}}$

• $\frac{3}{\sqrt{13}}$

The radius of the circle passing through the foci of the ellipse $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$ and having its centre at (0, 3) is

• 6

• 4

• 3

• 2

The equation of the circle passing through (2, 0) and (0, 4) and having the minimum radius, is

• x² + y² = 20

• x² + y² - 2x - 4y = 0

• x² + y² = 4

• x² + y² = 16

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

• ( - 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

• $\frac{3 \pm \sqrt{5}}{2}$

• $\frac{6 \pm \sqrt{5}}{2}$

• $\frac{9 \pm \sqrt{5}}{2}$

• $\frac{7 \pm \sqrt{5}}{2}$