The equation of a circle passing through origin and radius is a, is

• (x - a)² + (y - a)² = a²

• x² + y² = a²

• (x - a)² + y² = a²

• None of the above

Equation of tangent to the circle x² + y² - 2x - 2y + 1 = 0 perpendicular to y = x is given by

• $\mathrm{x} + \mathrm{y} \pm 1 = 0$

• $\mathrm{x} + \mathrm{y} = 2 \pm \sqrt{3}$

• $\mathrm{x} + \mathrm{y} \pm 3 = 0$

• None of these

243.

The locus of centre of circles which cuts orthogonally the circle x² + y² - 4x + 8 = 0 and touches x + 1 = 0, is

• y² + 6x + 7 = 0

• x² + y² + 2x + 3 = 0

• x² + 3y + 4 = 0

• None of the above

The condition for the line lx + my + n = 0 to be a normal to $\frac{{\mathrm{x}}^2}{25} + \frac{{\mathrm{y}}^2}{9} = 1$ is

• $\frac{{\mathrm{l}}^2}{9} + \frac{{\mathrm{m}}^2}{\mathrm{l}25} = \frac{{\mathrm{n}}^2}{256}$

• $\frac{9}{{\mathrm{m}}^2} + \frac{25}{{\mathrm{l}}^2} = \frac{256}{{\mathrm{n}}^2}$

• $\frac{{\mathrm{l}}^2}{9} - \frac{{\mathrm{m}}^2}{\mathrm{l}25} = \frac{{\mathrm{n}}^2}{256}$

• None of these

The radical centre of the system of circles

            x² + y² + 4x + 7 = 0,

2(x² + y²) + 3x + 5y + 9 = 0

and               x² + y² + y = 0 is

• (- 2, - 1)

• (1, - 2)

• (- 1, - 2)

• None of these

The point on the straight line y = 2x + 11 which is nearest to the circle 16(x² + y) + 32x - 8y - 50 = 0, is

• $\left(\frac{9}{2}, 2\right)$

• $\left(\frac{9}{2}, - 2\right)$

• $\left(- \frac{9}{2}, 2\right)$

• $\left(- \frac{9}{2}, - 2\right)$

The locus of the extrimities of the latusrectum of the family of ellipses b²x² + y² = a²b² having a given major axis, is

• ${\mathrm{x}}^2 \pm \mathrm{ay} = {\mathrm{a}}^2$

• ${\mathrm{y}}^2 \pm \mathrm{bx} = {\mathrm{a}}^2$

• ${\mathrm{x}}^2 \pm \mathrm{by} = {\mathrm{a}}^2$

• ${\mathrm{y}}^2 \pm \mathrm{ax} = {\mathrm{b}}^2$

The number of common tangents to two circles x² + y² = 4 and x² + y² - 8x + 12 = 0 is

• 1

• 2

• 3

• 4

If the tangent at the point $\left(2\sec \left(\mathrm{\theta }\right), 3\tan \left(\mathrm{\theta }\right)\right)$ of the hyperbola $\frac{{\mathrm{x}}^2}{4} - \frac{{\mathrm{y}}^2}{9} = 1$ is parallel to 3x - y + 4 = 0, then the value of $\mathrm{\theta }$ is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

If an equilateral triangle is inscribed in the circle x² + y² = a², the lenth of its each side is

• $\sqrt{2}\mathrm{a}$

• $\sqrt{3}\mathrm{a}$

• $\frac{\sqrt{3}}{2}\mathrm{a}$

• $\frac{1}{\sqrt{3}}\mathrm{a}$