431.

The number of common tangents that can be drawn to the circles x² + y² - 4x - 6y - 3 = 0 and x² + y² + 2x + 2y + 1 = 0 is

• 1

• 2

• 3

• 4

If two circles (x - 1)² + (y - 3)² = r² and x² + y² - 8x + 2y + 8 = 0 intersect in two distinct points, then

• 2 < r < 8

• r < 2

• r = 2

• r > 2

The equation of the tangent at the vertex of the parabola x² + 4x + 2y = 0, is

• x = - 2

• x = 2

• y = - 2

• y = 2

The distance between the directrices of the ellipse $\frac{{\mathrm{x}}^2}{4} + \frac{{\mathrm{y}}^2}{9} = 1$ is

• $\frac{9}{\sqrt{5}}$

• $\frac{18}{\sqrt{5}}$

• $\frac{24}{\sqrt{5}}$

• None of these

The equation of the normal to the hyperbola x² - 16y² - 2x - 64y - 72 = 0 at the point (- 4, - 3) is

• 5x + 16y + 79 = 0

• 16x + 5y + 97 = 0

• 16x + 5y + 79 = 0

• 5x + 16y + 97 = 0

The centre of the circle which circumscribes the square formed by x² - 8x + 12 = 0 and y² - 14y + 45 = 0 is

• (4, 5)

• (3, 4)

• (9, 5)

• (4, 7)

If e₁ and e₂ are the eccentricities of the hyperbolas $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{and} \frac{{\displaystyle {\mathrm{x}}^2}}{{\displaystyle {\mathrm{a}}^2}} - \frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}} = - 1$, then the value of $\frac{1}{{{\mathrm{e}}_1}^2} + \frac{1}{{{\mathrm{e}}_2}^2}$ is

• 3

• 2

• 1

• $\frac{1}{2}$

The point of contact of 3x + 4y + 7 = 0 and x² + y² - 4x - 6y -12 = 0 is

• (1, 1)

• (- 1, 1)

• (1, - 1)

• ( -1, - 1)

The equation x² - 7xy + 12y² = 0 represents a

• circle

• pair of parallel straight lines

• pair of perpendicular straight lines

• pair of non-perpendicular straight lines

The locus of z given by $\left|\frac{\mathrm{z} - 1}{\mathrm{z} +\hspace{0.17em}1}\right|$ = 1 is

• a parabola

• an ellipse

• a circle

• a straight line