The distance between the directrices of a rectangular hyperbola x² - y² = a² is 10 units, then distance between its foci is

• $10\sqrt{2}$

• 5

• $5\sqrt{2}$

• 20

The equation of the tangents of hyperbola 3x² - 4y² = 12 which cuts equal intercepts from both the axes, are

• $\mathrm{y} + \mathrm{x} = \pm 1$

• $\mathrm{x} - \mathrm{y} = \pm 1$

• $\mathrm{y} - \mathrm{x} = \pm 1$

• 4y - 3x = 0

Equation of the tangent to the hyperbola 2x² - 3y² = 6. Which is parallel to the line y - 3x - 4 = 0 is

• y = 3x + 8

• y = 3x - 8

• y = 3x + 2

• None of these

214.

The equation of circle which touches the axes and the line and whose centre lies inthe first quadrant is x² + y² - 2cx - 2cy + c² = 0. Then, c is equal to

• 1

• 2

• 3

• 6

The equation of the parabola having the focus at the point (3, - 1) and the vertex at (2, - 1)is

• y² - 4x - 2y + 9 = 0

• y² + 4x + 2y - 9 = 0

• y² - 4x + 2y + 9 = 0

• y² + 4x - 2y + 9 = 0

Find the equation of tangents to the ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ which cut off equal intercepts on the axes.

• $\mathrm{y} = \sqrt{3}\mathrm{x} \pm \sqrt{3{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\mathrm{y} = \pm \mathrm{x} \mp \sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\mathrm{y} = \sqrt{3}\mathrm{x} \pm \sqrt{{\mathrm{a}}^2 + 3{\mathrm{b}}^2}$

• None of the above

The locus ofthe point of intersection of the lines $\mathrm{xcos}\left(\mathrm{\alpha }\right) + \mathrm{ysin}\left(\mathrm{\alpha }\right) = \mathrm{p}$ and $\mathrm{xsin}\left(\mathrm{\alpha }\right) - \mathrm{ycos}\left(\mathrm{\alpha }\right) = \mathrm{q}$ ($\mathrm{\alpha }$ is a variable) will be

• a circle

• a staright line

• a parabola

• an ellipse

The locus of the mid points of the chords of a circle which subtend a right angle at its centre (equation ofthe circle is x² + y² = a²)will be

• x² + y² = 3a²

• x² + y² = $\frac{{\mathrm{a}}^2}{3}$

• 2(x² + y²) = a²

• 4(x² + y²) = a²

If the line 3x - 2y + p = 0 is normal to the circle x² + y² = 2x - 4y - 1, then p will be

• - 5

• 7

• - 7

• 5

If the two circles x² + y² = r² and x² + y² - 10x + 16 = 0 intersect at two real points, then

• 1 < r < 7

• 3 < r < 10

• 2 < r < 9

• 2 < r < 8