Mathematics : Conic Section

If the line lx + my = 1 is a normal to the hyperbola

$\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\displaystyle {\mathrm{y}}^2}}{{\mathrm{b}}^2} = 1, \mathrm{then} \frac{{\mathrm{a}}^2}{{\mathrm{l}}^2} - \frac{{\mathrm{b}}^2}{{\mathrm{m}}^2} \mathrm{is} \mathrm{equal} \mathrm{to}$

• a² - b²

• a² + b²

• (a² + b²)²

• (a² - b²)²

If the lines 2x - 3y = 5 and 3x - 4y = 7 are two diameters of a circle of radius 7, then the equation of the circle is

• x² + y² + 2x - 4y - 47 = 0

• x² + y² = 49

• x² + y² - 2x + 2y - 47 = 0

• x² + y² = 17

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

x² + y² - 4x - 6y + 9 = 0, is

• $\left(1, \frac{1}{2}\right)$

• $\left(2, 1\right)$

• $\left(0, 1\right)$

• $\left(1, 0\right)$

If $\mathrm{\theta }$ is the angle between the tangents from(- 1, 0) to the circle x² + y² - 5x + 4y - 2 = 0, then $\mathrm{\theta }$ is equal to

• $2{\tan }^{-1}\left(\frac{7}{4}\right)$

• ${\tan }^{-1}\left(\frac{7}{4}\right)$

• $2{\cos }^{-1}\left(\frac{7}{4}\right)$

• ${\cot }^{-1}\left(\frac{7}{4}\right)$

If 2x + 3y + 12 = 0 and x - y + 4$\mathrm{\lambda }$ = 0 are conjugate with respect to the parabola y = 8x, then $\mathrm{\lambda }$ is equal to

• 2

• - 2

• 3

• - 3

For an ellipse with eccentricity $\frac{1}{2}$ the centre is at the origin. If one directrix is x = 4, then the equation of the ellipse is

• $3{\mathrm{x}}^2 + 4{\mathrm{y}}^2 = 1$

• $3{\mathrm{x}}^2 + 4{\mathrm{y}}^2 = 12$

• $4{\mathrm{x}}^2 + 3{\mathrm{y}}^2 = 1$

• $4{\mathrm{x}}^2 + 3{\mathrm{y}}^2 = 12$

The distance between the foci of the hyperbola x² - 3y² - 4x - 6y - 11 = 0 is

• 4

• 6

• 8

• 10

The radius of the circle with the polar equation

r² - 8r($\sqrt{3}$cos$\mathrm{\theta }$ + sin$\mathrm{\theta }$) + 15 = 0 is

• 8

• 7

• 6

• 5

The area (in square unit) of the circle which touches the lines 4x + 3y = 15 and 4x + 3y = 5 is

• 4$\mathrm{\pi }$

• $\mathrm{3\pi }$

• $\mathrm{2\pi }$

• $\mathrm{\pi }$

The equations of the circle which pass through the origin and makes intercepts of length 4 and 8 on the x and y-axes respectively are

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 4\mathrm{x} \pm 8\mathrm{y} = 0$

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm 2\mathrm{x} \pm 4\mathrm{y} = 0$

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

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 \pm \mathrm{x} \pm \mathrm{y} = 0$