31.

The area (in square unit) of the triangle formed by x + y + 1 = 0 and the pair of straight lines x² - 3xy + 2y² = 0 is

• $\frac{7}{12}$

• $\frac{5}{12}$

• $\frac{1}{12}$

• $\frac{1}{6}$

The pairs of straight lines x² - 3xy + 2y² = 0 and x - 3xy + 2y² + x - 2 = 0 form a

• Square but not rhombus

• rhombus

• Parallelogram

• Rectangle but not a square

The equations of the circle which pass through the origin and makes intercepts of length 4 and 8 on the x and y-axes respectively are

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 4\mathrm{x} \pm 8\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 2\mathrm{x} \pm 4\mathrm{y} = 0$

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

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm \mathrm{x} \pm \mathrm{y} = 0$

The locus of centre of a circle which passes through the origin and cuts off a length of 4 unit from the line x = 3 is

• y² + 6x = 0

• y² + 6x = 13

• y² + 6x = 10

• x² + 6y = 13

The diameters of a circle are along 2x +y - 7 and x + 3y - 11 = 0. Then, the equation of this circle, which also passes through (5, 7) is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 6\mathrm{y} - 16 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 6\mathrm{y} - 20 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 4\mathrm{x} - 6\mathrm{y} - 12 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 4\mathrm{x} + 6\mathrm{y} - 12 = 0$

$$\begin{gathered} \mathrm{The} \mathrm{point} \left(3, - 4\right) \mathrm{lies} \mathrm{on} \mathrm{both} \mathrm{the} \mathrm{circles} \\ {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 - 2\mathrm{x} + 8\mathrm{y} +\hspace{0.17em}13 = 0 \mathrm{and} {\mathrm{x}}^2 +\hspace{0.17em}{\mathrm{y}}^2 -\hspace{0.17em}4\mathrm{x} +\hspace{0.17em}6\mathrm{y} + 11 = 0, \\ \mathrm{Then}, \mathrm{the} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{circles} \mathrm{is} \end{gathered}$$

• $60°$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$

• ${\tan }^{-1}\left(\frac{3}{5}\right)$

• $135°$

• 3x² + 3y² - 8x - 13y = 0

• 3x² + 3y² - 8x + 29y = 0

• 3x² + 3y² + 8x + 29y = 0

• 3x² + 3y² - 8x - 29y = 0

The number of normals drawn to the parabola y² = 4x from the point (1, 0) is

• 0

• 1

• 2

• 3

If the distance between foci of an ellipse is 6 and the length of the minor axis is 8, then the eccentricity is

• $\frac{1}{\sqrt{5}}$

• $\frac{1}{2}$

• $\frac{3}{5}$

• $\frac{4}{5}$

If the circle x² + y² = a² intersects the hyperbola xy = c² in four points (x_i, y_i), for i = 1, 2, 3 and 4, then y₁ + y₂ + y₃ + y₄ equals

• 0

• c

• a

• c⁴