Mathematics : Conic Section

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 :

• $\sqrt{2}$

• $\sqrt{3}$

• $\sqrt{\frac{3}{2}}$

• $2\sqrt{3}$

The sides of the rectangle of greatest area that can be inscribed in the ellipse x² + 4y² = 64 are :

• $\left(6\sqrt{2}, 4\sqrt{2}\right)$

• $\left(8\sqrt{2}, 4\sqrt{2}\right)$

• $\left(8\sqrt{2}, 8\sqrt{2}\right)$

• $\left(16\sqrt{2}, 4\sqrt{2}\right)$

The polar equation of the circle with centre $\left(2, \frac{\mathrm{\pi }}{2}\right)$ are radius 3 units is:

• ${\mathrm{r}}^2 + 4\mathrm{rcos}\left(\mathrm{\theta }\right) = 5$

• ${\mathrm{r}}^2 + 4\mathrm{rsin}\left(\mathrm{\theta }\right) = 5$

• ${\mathrm{r}}^2 - 4\mathrm{rsin}\left(\mathrm{\theta }\right) = 5$

• ${\mathrm{r}}^2 - 4\mathrm{rcos}\left(\mathrm{\theta }\right) = 5$

The equation of the circle of radius 3 that lies in the fourth quadrant and touching the lines x = 0 and y = 0 is

• x² + y² - 6x + 6y + 9 = 0

• x² + y² - 6x - 6y + 9 = 0

• x² + y² + 6x - 6y + 9 = 0

• x² + y² + 6x + 6y + 9 = 0

The inverse point of (1, 2) with respect to the circle

x² + y² - 4x - 6y + 9 = 0 is

• (0, 0)

• (1, 0)

• (0, 1)

• (1, 1)

The condition for the coaxial system ^x2 + y² + 2$\mathrm{\lambda }$x + c = 0, where $\mathrm{\lambda }$, is a parameter and c is a constant, to have distinct limiting points, is

• c = 0

• c $<$ 0

• c = - 1

• c $>$ 0 

For the parabola y² + 6y - 2x + 5 = 0

(I) The vertex is (- 2, - 3)

(II) The directrix is y + 3 = 0

Which of the following is correct ?

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

The value of k, If (1, 2), (k, - 1) are conjugate points with respect to the ellipse 2x² + 3y² = 6 is

• 2

• 4

• 6

• 8