Conic Section

The equation of the circle which passes through the points of intersection of the circles x² + y² - 6x = 0 and x² + y² - 6y = 0 and has its centre at $\left(\frac{3}{2}, \frac{{\displaystyle 3}}{{\displaystyle 2}}\right)$, is

• x² + y² + 3x + 3y + 9 = 0

• x² + y² + 3x + 3y = 0

• x² + y² - 3x - 3y = 0

• x² + y² - 3x - 3y + 9 = 0

If 2y = x and 3y + 4x = 0 are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is

• $\sqrt{\frac{2}{3}}$

• $\sqrt{\frac{2}{5}}$

• $\sqrt{\frac{1}{3}}$

• $\sqrt{\frac{1}{2}}$

If t is a parameter, then x = $\mathrm{a}\left(\mathrm{t} + \frac{1}{\mathrm{t}}\right)$, y = $\mathrm{b}\left(\mathrm{t} - \frac{1}{\mathrm{t}}\right)$ represents

• an ellipse

• a circle

• a pair of straight lines

• a hyperbola

The equation (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0 represents a circle whose centre is

• $\left(\frac{{\mathrm{x}}_1 - {\mathrm{x}}_2}{2}, \frac{{\displaystyle {\mathrm{y}}_1 - {\mathrm{y}}_2}}{{\displaystyle 2}}\right)$

• $\left(\frac{{\mathrm{x}}_1 + {\mathrm{x}}_2}{2}, \frac{{\displaystyle {\mathrm{y}}_1 + {\mathrm{y}}_2}}{{\displaystyle 2}}\right)$

• (x₁, y₁)

• (x₂, y₂)

The circles x² + y² + 6x + 6y = 0 and x² + y² - 12x - 12y = 0

• cut orthogonally

• touch each other internally

• intersect in two points

• touch each other externally

The two parabolas x² = 4y and y² = 4x meet in two distinct points. One of these is the origin and the other is

• (2, 2)

• (4, - 4)

• (4, 4)

• (- 2, 2)

The vertex of the parabola x² + 2y = 8x - 7 is

• $\left(\frac{9}{2}, 0\right)$

• $\left(4, \frac{9}{2}\right)$

• $\left(2, \frac{9}{2}\right)$

• $\left(4, \frac{7}{2}\right)$

If P(at², 2at) be one end of a focal chord of the parabola y² = 4ax, then the length of the chord is

• a${\left(\mathrm{t} - \frac{1}{\mathrm{t}}\right)}^2$

• a$\left(\mathrm{t} - \frac{1}{\mathrm{t}}\right)$

• a$\left(\mathrm{t} + \frac{1}{\mathrm{t}}\right)$

• a${\left(\mathrm{t} + \frac{1}{\mathrm{t}}\right)}^2$

The length of the common chord of the parabolas y² = x and x² = y is

• $2\sqrt{2}$

• 1

• $\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

The equation of the ellipse having vertices at (± 5, 0) and foci (± 4, 0) is

• $\frac{{\mathrm{x}}^2}{25} + \frac{{\mathrm{y}}^2}{16} = 1$

• 9x² + 25y² = 225

• $\frac{{\mathrm{x}}^2}{9} + \frac{{\mathrm{y}}^2}{25} = 1$

• 4x² + 5y² = 20