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

• $\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

137.

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

The area included between the parabolas y² = 4x and x² = 4y is

• $\frac{8}{3}$ sq unit

• 8 sq unit

• $\frac{16}{3}$ sq unit

• 12 sq unit

The locus of the centres of the circles which touch both the axes is given by

• x² - y² = 0

• x² + y² = 0

• x² - y² = 1

• x² + y² = 1

The latusrectum of an ellipse is equal to one-half of its minor axis. The eccentricity of the ellipse is

• $\frac{1}{\sqrt{6}}$

• $\frac{\sqrt{3}}{2}$

• $\frac{\sqrt{3}}{4}$

• $\frac{1}{2}$