The distance between the focii of the ellipse
x = 3cos$\left(\mathrm{\theta }\right)$, y = 4sin$\left(\mathrm{\theta }\right)$ is

• $2\sqrt{7}$

• $7\sqrt{2}$

• $\sqrt{7}$

• $3\sqrt{7}$

The equations of the latus rectum of the ellipse
9x² + 25y² - 36x + 50y - 164 = 0 are

• x - 4 = 0, x + 2 = 0

• x - 6 = 0, x + 2 = 0

• x + 6 = 0, x - 2 = 0

• x + 4 = 0, x + 5 = 0

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 equation of the common tangent drawn to the curves y = 8x and xy = - 1 is

• y = 2x + 1

• 2y = x + 6

• y = x + 2

• 3y = 8x + 2

The area included between the parabola $\mathrm{y} = \frac{{\mathrm{x}}^2}{4\mathrm{a}} \mathrm{and} \mathrm{the} \mathrm{curve} \mathrm{y} = \frac{8{\mathrm{a}}^3}{\left({\mathrm{x}}^2 + 4{\mathrm{a}}^2\right)} \mathrm{is}$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } + \frac{2}{3}\right)$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } - \frac{8}{3}\right)$

• ${\mathrm{a}}^2\left(\mathrm{\pi } + \frac{4}{3}\right)$

• ${\mathrm{a}}^2\left(2\mathrm{\pi } - \frac{4}{3}\right)$

If a circle with radius 2.5 units passes through the points (2, 3) and (5, 7), then its centre is

• (1 5, 2)

• (7, 10)

• (3, 4)

• (3 5, 5)

The circumcentre of the triangle formed by the points (1, 2, 3) (3, - 1, 5), (4, 0, - 3) is

• (1, 1, 1)

• (2, 2, 2)

• (3, 3, 3)

• $\left(\frac{7}{2}, \frac{ - 1}{2}, 1\right)$

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

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

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

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

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

409.

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

$\frac{\mathrm{x} - 1}{3\mathrm{x} + 4} < \frac{\mathrm{x} - 3}{3\mathrm{x} - 2} \mathrm{holds}, \mathrm{for} \mathrm{all} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{internal}$

• $\left(\frac{ - 4}{3}, \frac{2}{3}\right)$

• $\left(\infty , \frac{ - 5}{4}\right)$

• $\left(\frac{3}{3}, \infty \right)$

• $\left( - \infty , \frac{ - 5}{4}\right) \cup \left(\frac{3}{4}, - \infty \right)$