Conic Section

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

• $\frac{3 \pm \sqrt{5}}{2}$

• $\frac{6 \pm \sqrt{5}}{2}$

• $\frac{9 \pm \sqrt{5}}{2}$

• $\frac{7 \pm \sqrt{5}}{2}$

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 lines y = 2x + $\sqrt{76}$ and 2y + x = 8 touch the ellipse x² + y² = 1. If the point of $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{12} = 1$ intersection of these two lines lie on a circle, whose centre coincides with the centre of that ellipse, then the equation of that circle is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = 28$

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

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

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = {\left(4 + \sqrt{8}\right)}^2$

If lx + my = 1 is a normal to the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$, then a²m² - b²l² = ?

• $\frac{{\mathrm{m}}^2}{{\mathrm{l}}^2}{\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2$

• l² + m²(a² + b²)²

• $\frac{{\mathrm{l}}^2}{{\mathrm{m}}^2}{\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right)}^2$

• l²m²(a² + b²)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

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

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

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

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

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

• a triangle

• an ellipse

• a circle

• a square