The mid-point of a chord of the ellipse x² + 4y² - 2x + 20y = 0 is (2, - 4). The equation of the chord is

• x - 6y = 26

• x + 6y = 26

• 6x - y = 26

• 6x + y = 26

If the focus of the ellipse $\frac{{\mathrm{x}}^2}{25} + \frac{{\mathrm{y}}^2}{16} = 1$ and the hyperbola $\frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ coincide, then b² =?

• 4

• 5

• 8

• 9

If a = 9 is a chord of contact of the hyperbola x² - y² = 9, then the equation of the tangent at one of the points of contact is 

• $\mathrm{x} + \sqrt{3}\mathrm{y} + 2 = 0$

• $3\mathrm{x} + 2\sqrt{2}\mathrm{y} - 3 = 0$

• $3\mathrm{x} - \sqrt{2}\mathrm{y} + 6 = 0$

• $\mathrm{x} - \sqrt{3}\mathrm{y} + 2 = 0$

The point at which the circles x² + y² - 4x - 4y + 7 = 0 and x² + y² - 12x - 10y + 45 = 0 touch each other, is

• $\left(\frac{13}{5}, \frac{{\displaystyle 14}}{{\displaystyle 5}}\right)$

• $\left(\frac{2}{5}, \frac{{\displaystyle 5}}{{\displaystyle 6}}\right)$

• $\left(\frac{14}{5}, \frac{{\displaystyle 13}}{{\displaystyle 5}}\right)$

• $\left(\frac{12}{5}, 2 + \frac{{\displaystyle \sqrt{21}}}{{\displaystyle 5}}\right)$

The length of the common chord of the two circles x² + y² - 4y = 0 and x² + y² - 8x - 4y + 11 = 0, is

• $\frac{\sqrt{145}}{4} \mathrm{cm}$

• $\frac{\sqrt{11}}{2} \mathrm{cm}$

• $\sqrt{135} \mathrm{cm}$

• $\frac{\sqrt{135}}{4}$

The locus of the centre of the circle, which cuts the circle x² + y² - 20 + 4 = 0 orthogonally and touches the line x = 2, is

• x² = 16y

• y² = 4x

• y² = 16x

• x² = 4y

337.

If a normal chord at a point t on the parabola y² = 4ax subtends a right angle at the vertex, then t equals to

• 1

• $\sqrt{2}$

• 2

• $\sqrt{3}$

The slopes of the focal chords of the parabola y² = 32x, which are tangents to the circle x² + y² = 4, are

• $\frac{1}{2}, - \frac{1}{2}$

• $\frac{1}{\sqrt{3}}, \frac{- 1}{\sqrt{3}}$

• $\frac{1}{\sqrt{15}}, \frac{- 1}{\sqrt{15}}$

• $\frac{2}{\sqrt{5}}, \frac{- 2}{\sqrt{5}}$

If tangents are drawn from any point on the circle x² + y² = 25 to the ellipse $\frac{{\mathrm{x}}^2}{16} + \frac{{\mathrm{y}}^2}{9} = 1$,  then the angle between the tangents is

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

An ellipse passing through has $\left(4\sqrt{2}, 2\sqrt{6}\right)$ foci at (- 4, 0) and (4, 0). Then, its eccentricity

• $\sqrt{2}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{\sqrt{3}}$