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 x² - 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 lengths 4 and 8 on the x and y -axes respectively are

• x² + y² ± 4x ± 8y = 0

• x² + y² ± 2x ± 4y = 0

• x² + y² ± 8x ± 16y = 0

• x² + y² ± x ± y = 0

The point (3 -4) lies on both the circles x² + y² - 2x + 8y + 13 = 0 and x² + y² - 4x + 6y + 11 = 0. Then, the angle between the circles is

• $60°$

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

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

• 135$°$

The equation of the circle which passes through the origin and cuts orthogonally each of the circles x² + y² - 6x + 8 = 0 and x² + y² - 2x - 2y = 7 is

• 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

17.

If the circle x² + y² = a 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

The mid point of the chord 4x - 3y = 5 of the hyperbola 2x² - 3y² = 12 is

• $\left(0, - \frac{5}{3}\right)$

• (2, 1)

• $\left(\frac{5}{4}, 0\right)$

• $\left(\frac{11}{4}, 2\right)$

The perimeter of the triangle with vertices at (1, 0, 0), (0, 1, 0) and (0, 0, 1) is

• 3

• 2

• 2$\sqrt{2}$

• 3$\sqrt{2}$

If a line in the space makes angle $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ with the coordinate axes, then

$\cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right) + {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) + {\sin }^2\left(\mathrm{\gamma }\right)$ equals

• - 1

• 0

• 1

• 2