The product of the perpendicular distances from any point on the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1$ to its asymtotes is

• $\frac{{\mathrm{a}}^2{\mathrm{b}}^2}{{\mathrm{a}}^2 - {\mathrm{b}}^2}$

• $\frac{{\mathrm{a}}^2{\mathrm{b}}^2}{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\frac{{\mathrm{a}}^2 + {\mathrm{b}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}$

• $\frac{{\mathrm{a}}^2 - {\mathrm{b}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}$

If the lines 2x + 3y +12 = 0, x - yy + k = 0 are conjugate with respect to the parabola y² = 8x, then k is equal to

• 10

• $\frac{7}{2}$

• - 12

• - 2

Find the equation to the parabola, whose axis parallel to they-axis and which passes through the points (0, 4), (1, 9) and (4, 5) is

• y = - x² + x + 4

• y = - x² + x + 1

• $\mathrm{y} = \frac{- 19}{12}{\mathrm{x}}^2 + \frac{79}{12}\mathrm{x} + 4$

• $\mathrm{y} = \frac{- 19}{12}{\mathrm{x}}^2 + \frac{89}{12}\mathrm{x} + 4$

If the line y = 2x + c is a tangent to the circle x² + y² = 5, then a value of

• 2

• 3

• 4

• 5

A line segment AM = a moves in the XOY plane such that AM is parallel to the X-axis. If A moves along the circle x² + y² = a², then the locus of M is

• x² + y² = 4a²

• x² + y² = 2ax

• x² + y² = 2ay

• x² + y² = 2ax + 2ay

If a chord of the parabola y = 4x passes through its focus and makes an angle 0 with the X-axis, then its length is

• $4{\cos }^2\left(\mathrm{\theta }\right)$

• $4{\sin }^2\left(\mathrm{\theta }\right)$

• $4{\csc }^2\left(\mathrm{\theta }\right)$

• $4{\sec }^2\left(\mathrm{\theta }\right)$

317.

If the straight line y = mx + c is parallel to the axis of the parabola y = bx and intersects the parabola at $\left(\frac{{\mathrm{c}}^2}{8}, \mathrm{c}\right)$, then the length of the latus rectum is

• 2

• 3

• 4

• 8

The eccentricity of the ellipse x² + 4y² + 2x + 16y + 13 = 0 is

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

The angle between the asymptotes of the hyperbola x² - 3y² = 3 is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

If the area of the triangle formed by the pair of lines 8x² - 6xy + y² = 0 and the line 2x + 3y = a is 7, then a is equal to

• 14

• $14\sqrt{2}$

• $28\sqrt{2}$

• 28