If y = 3x is a tangent to a circle with centre (1, 1), then the other tangent drawn through (0, 0) to the circle is

• 3y = x

• y = - 3x

• y = 2x

• y = - 2x

The line among the following which touches the parabola y = 4ax, is

• x + my + am² = 0

• x - my + am² = 0

• x + my - am² = 0

• y + mx + am² = 0

Which of the following equations gives a circle ?

• $\mathrm{r} = 2\sin \left(\mathrm{\theta }\right)$

• ${\mathrm{r}}^2\cos \left(2\mathrm{\theta }\right) = 1$

• $\mathrm{r}\left(4\cos \left(\mathrm{\theta }\right) + 5\sin \left(\mathrm{\theta }\right)\right) = 3$

• $5 = \mathrm{r}\left(1 + \sqrt{2}\cos \left(\mathrm{\theta }\right)\right)$

Let O be the origin and A be a point on the curve y = 4x. Then the locus of the mid point of OA is :

• x² = 4y

• x² = 2y

• y² = 16x

• y² = 2x

The number of common tangents to the two circles x² + y - 8x + 2y = 0 and x² + y² - 2x - 16y + 25 = 0 is :

• 1

• 2

• 3

• 4

Observe the following statements :

I. The circle x² + y² - 6x - 4y - 7 = 0 touches y-axis.

II. The circle x² + y² + 6x + 4y - 7 = 0 touches

x-axis. Which of the following is a correct statement ?

• Both I and II are true

• Neither I nor II is true

• I is true, II is false

• I is false, II is true

The length of the tangent drawn to the circle x² + y² - 2x + 4y - 11 = 0

• 1

• 2

• 3

• 4

If b and c are the lengths of the segments of any focal chord of a parabola y² = 4ax, then the length of the semi-latus rectum is

• $\frac{\mathrm{bc}}{\mathrm{b} + \mathrm{c}}$

• $\sqrt{\mathrm{bc}}$

• $\frac{\mathrm{b} + \mathrm{c}}{2}$

• $\frac{2\mathrm{bc}}{\mathrm{b} + \mathrm{c}}$

Equations of the latus rectum of the ellipse

9x² + 4y² - 18x - 8y - 23 = 0 are : 

• $\mathrm{y} = \pm \sqrt{5}$

• $\mathrm{x} = \pm \sqrt{5}$

• $\mathrm{y} = 1 \pm \sqrt{5}$

• $\mathrm{x} = - 1 \pm \sqrt{5}$

If the eccentricity of a hyperbola is $\sqrt{3}$; then the eccentricity of its conjugate hyperbola is :

• $\sqrt{2}$

• $\sqrt{3}$

• $\sqrt{\frac{3}{2}}$

• $2\sqrt{3}$