The equation of the circle of radius 5 and touching the co-ordinate axes in third quadrant is

• (x - 5)²+ (y + 5)² = 25 

• (x + 5)² + (y + 5)² = 25

• (x + 4)² + (y + 4)² = 25

• (x + 6)² + (y + 6)² = 25

The four distinct points (0, 0), (2, 0), (0, - 2)and (k, - 2) are concyclic, if k is equal to

• 3

• 1

• - 2

• 2

13.

A line is at a constant distance c from the origin and meets the coordinate axes in A and B. The locus of the centre of the circle passing through O, A, B is

• x² + y² = c²

• x² + y² = 2c²

• x² + y² = 3c²

• x² + y² = 4c²

The line y = mx + c intercepts the circle x² + y² = r² in two distinct points, if

• $- \mathrm{r}\sqrt{1 + {\mathrm{m}}^2} < \mathrm{c} < \mathrm{r}\sqrt{1 + {\mathrm{m}}^2}$

• $\mathrm{c} < - \mathrm{r}\sqrt{1 + {\mathrm{m}}^2}$

• $\mathrm{c} < \mathrm{r}\sqrt{1 + {\mathrm{m}}^2}$

• None of the above

The equation of the parabola with the focus (3, 0) and the directrix x + 3 = 0, is

• ${\mathrm{y}}^2 = 3\mathrm{x}$

• ${\mathrm{y}}^2 = 6\mathrm{x}$

• ${\mathrm{y}}^2 = 12\mathrm{x}$

• ${\mathrm{y}}^2 = 2\mathrm{x}$

If e and e' are the eccentricities of the ellipse 5x² + 9y² = 45 and the hyperbola 5 - 4y = 45 respectively, then ee' is equal to

• 1

• 4

• 5

• 9

The pole of the straight line x + 4y = 4 with respect to the ellipse x² + 4y² = 4 is

• (1, 1)

• (1, 4)

• (4, 1)

• (4, 4)

Locus of the poles of focal chord of a parabola is

• the axis

• a focal chord

• the directrix

• the tangent at the vertex

The equation $\frac{1}{\mathrm{r}} = \frac{1}{8} + \frac{3}{8}\cos \left(\mathrm{\theta }\right)$ represents

• a parabola

• an ellipse

• a hyperbola

• a rectangular hyperbola

$\underset{\mathrm{x} \to 0}{\lim } \frac{4^{\mathrm{x}} - 9^{\mathrm{x}}}{\mathrm{x}\left(4^{\mathrm{x}} + 9^{\mathrm{x}}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\log \left(\frac{2}{3}\right)$

• $\log \left(\frac{3}{2}\right)$

• $\frac{1}{2}\log \left(\frac{2}{3}\right)$

• $\frac{1}{2}\log \left(\frac{3}{2}\right)$