Let A(- 1, 0) and B(2, 0) be two points. A point M moves in the plane in such a way that $\angle \mathrm{MBA} = 2\angle \mathrm{MAB}$. Then, the point M moves along

• a straight line

• a parabola

• an ellipse

• a hyperbola

The area of the figure bounded by the parabolas x = - 2y ² and x = 1- 3y² is

• $\frac{4}{3}$ sq. units

• $\frac{2}{3}$ sq. units

• $\frac{3}{7}$ sq. units

• $\frac{6}{7}$sq. units

Tangents are drawn to the ellipse $\frac{{\mathrm{x}}^2}{9} + \frac{{\mathrm{y}}^2}{5} = 1$ at the ends of both latusrectum. The area of the quadrilateral, so formed is

• 27 sq. units

• $\frac{13}{2}$ sq. units

• $\frac{15}{4}$ sq. units

• 45 sq. units

If the tangent to y² = 4ax at the point (at², 2at) where $\left|\mathrm{t}\right|$ > 1 is a normal to x² - y² = a² at the point ($\mathrm{a} \sec \left(\mathrm{\theta }\right), \mathrm{a} \tan \left(\mathrm{\theta }\right)$), then

• $\mathrm{t} = - \csc \left(\mathrm{\theta }\right)$

• $\mathrm{t} = - \sec \left(\mathrm{\theta }\right)$

• $\mathrm{t} = 2\tan \left(\mathrm{\theta }\right)$

• $\mathrm{t} = 2\cot \left(\mathrm{\theta }\right)$

55.

The focus of the conic x- 6x + 4y + 1 = 0 is

• (2, 3)

• (3, 2)

• (3, 1)

• (1, 4)

The line y = x + $\mathrm{\lambda }$ is tangent to the ellipse 2x² + 3y² = 1. Then, $\mathrm{\lambda }$ is

• - 2

• 1

• $\sqrt{\frac{5}{6}}$

• $\sqrt{\frac{2}{3}}$

The equation of a line parallel to the line 3x + 4y= 0 and touching the circle x² + y² = 9 in the first quadrant, is

• 3x +4y = 15

• 3x +4y = 45

• 3x +4y = 9

• 3x +4y = 27

A line passing through the point of intersection of x + y = 4 and x - y = 2 makes an angle ${\tan }^{-1}\left(\frac{3}{4}\right)$ with the x-axis. It intersects the parabola y² = 4(x-3) at points $\left({\mathrm{x}}_1 , {\mathrm{y}}_1\right) \mathrm{and} \left({\mathrm{x}}_2, {\mathrm{y}}_2\right)$ respectively. Then, $\left|{\mathrm{x}}_1 - {\mathrm{x}}_2\right|$

• $\frac{16}{9}$

• $\frac{32}{9}$

• $\frac{40}{9}$

• $\frac{80}{9}$

The equation of auxiliary circle of the ellipse 16x² + 25y² + 32x - 100y = 284 is

• x² + y² + 2x - 4y - 20 = 0

• x² + y² + 2x - 4y = 0

• (x + 1)² + (y - 2)² = 400

• (x + 1)² + (y - 2)² = 225

If PQ is a double ordinate of the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{such} \mathrm{that} ∆\mathrm{OPQ}$ is equilateral. O being the centre. Then, the eccentricity e satisfies 

• $1 < \mathrm{e} < \frac{2}{\sqrt{3}}$

• e = $\frac{2}{\sqrt{2}}$

• e = $\frac{\sqrt{3}}{2}$

• $\mathrm{e} > \frac{2}{\sqrt{3}}$