Mathematics : Conic Section

The product of the lengths of perpendiculars drawn from any point on the hyperbola x² - 2y² - 2 = O to its asymptotes is

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{2}$

• 2

The equation of the parabola with focus (0, 0)and directrix x + y = 4 is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy} + 8\mathrm{x} +8\mathrm{y} -16 = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 - 2\mathrm{xy} + 8\mathrm{x} + 8\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 + 8\mathrm{x} + 8\mathrm{y} - 16= 0$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 + 8\mathrm{x} +8\mathrm{y} - 16= 0$

The number of circles that touch all the three lines x + y - 1 = 0, x - y - 1 = 0 and y + 1 = 0 is

• 2

• 3

• 4

• 1

If P₁, P₂, P₃ are the perimeters of the three circles 

x² + y² + 8x - 6y = 0, 4x² + 4y - 4x - 12y - 186 = 0 and x² + y - 6x + 6y - 9 = 0 respectively, then

• ${\mathrm{P}}_1 < {\mathrm{P}}_2 < {\mathrm{P}}_3$

• ${\mathrm{P}}_1 < {\mathrm{P}}_3 < {\mathrm{P}}_2$

• ${\mathrm{P}}_3 < {\mathrm{P}}_2 < {\mathrm{P}}_1$

• ${\mathrm{P}}_2 < {\mathrm{P}}_3 < {\mathrm{P}}_1$

If the line 3x - 2y + 6 = 0 meets X-axis and Y-axis, respectively at A and B, then the equation of the circle with radius AB and centre at A is

• x² + y² + 4x + 9 = 0

• x² + y² + 4x - 9 = 0

• x² + y² + 4x + 4 = 0

• x² + y² + 4x - 4 = 0

A line l meets the circle x² + y² = 61 in A, B and P(- 5, 6) is such that PA = PB = 10. Then,the equation of l is

• 5x + 6y + 11 = 0

• 5x - 6y - 11 = 0

• 5x - 6y + 11 = 0

• 5x - 6y + 11 = 0

If (1, a), (b, 2) are conjugate points with respect to the circle x² + y² = 25, then 4a + 2b is equal to

• 25

• 50

• 100

• 150

The eccentricity of the conic 36x² + 144y² - 36x - 96y -119 = 0 is

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{4}$

• $\frac{1}{\sqrt{3}}$

The polar equation $\cos \left(\mathrm{\theta }\right) + 7\sin \left(\mathrm{\theta }\right) = \frac{1}{\mathrm{r}}$ represents a

• circle

• parabola

• straight line

• hyperbola

The centre of the circle ${\mathrm{r}}^2 - 4\mathrm{r}\left(\cos \left(\mathrm{\theta }\right) + \sin \left(\mathrm{\theta }\right)\right) - 4 = 0$ in cartesian coordinates is

• (1, 1)

• (- 1, - 1)

• (2, 2)

• (- 2, - 2)