The pole of the straight line x + 4y = 4 with respect to the ellipse x² + 4y² = 4 is

• (1, 1)

• (1, 4)

• (4, 1)

• (4, 4)

Locus of the poles of focal chord of a parabola is

• the axis

• a focal chord

• the directrix

• the tangent at the vertex

The equation $\frac{1}{\mathrm{r}} = \frac{1}{8} + \frac{3}{8}\cos \left(\mathrm{\theta }\right)$ represents

• a parabola

• an ellipse

• a hyperbola

• a rectangular hyperbola

If the circle x² + y² + 6x - 2y + k = 0 bisects the circumference of the circle x² + y² + 2x - 6y - 15 = 0, then k is equal to:

• 21

• - 21

• 23

• - 23

If P is a point such that the ratio of the square of the lengths of the tangents from P to the circles x² + y² + 2x - 4y - 20 = 0 and x² + y² - 4x + 2y - 2y - 44 = 0 is 2 : 3, then the locus of P is a circle with centre :

• (7, - 8)

• (- 7, 8)

• (7, 8)

• (- 7, - 8)

If 5x - 12y + 10 = 0 and 12y - 5x + 16 = 0 are two tangents to a circle, then the radius of the circle is

• 1

• 2

• 4

• 6

The eccentricity of the ellipse 9x² + 5y² - 18x - 20y - 16 = 0, is:

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{2}$

• $2$

The product of the lengths of perpendiculars drawn from any point on the hyperbola x² - 2y² - 2 = O to its asymptotes is

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{2}$

• 2

The equation of the parabola with focus (0, 0)and directrix x + y = 4 is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy} + 8\mathrm{x} +8\mathrm{y} -16 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy} + 8\mathrm{x} + 8\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 8\mathrm{x} + 8\mathrm{y} - 16= 0$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 + 8\mathrm{x} +8\mathrm{y} - 16= 0$

The number of circles that touch all the three lines x + y - 1 = 0, x - y - 1 = 0 and y + 1 = 0 is

• 2

• 3

• 4

• 1