If the circle x² + y² + 2x + 3y + 1 = 0 cuts another circle x² + y² + 4x + 3y + 2 = 0 in A and B, then the equation of the circle with AB as a diameter is

• x² + y²  + x + 3y + 1 = 0

• 2x² + 2y²  + 2x + 6y + 1 = 0

• x² + y²  + x + 6y + 1 = 0

• 2x² + 2y²  + x + 3y + 1 = 0

The equation of the hyperbola which passes through the point (2, 3) and has the asymptotes 4x + 3y - 7 = 0 and x - 2y - 1 = 0 is

• 4x² + 5xy - 6y² - 11x + 11y + 50 = 0

• 4x² + 5xy - 6y² - 11x + 11y - 43 = 0

• 4x² - 5xy - 6y² - 11x + 11y + 57 = 0

• x² - 5xy - y² - 11x + 11y - 43 = 0

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

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

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

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

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

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

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$