Mathematics : Conic Section

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

If the circle x² + y² + 6x - 2y + k = 0 bisects the circumference of the circle x² + y² + 2x - 6y - 15 = 0, then k is equal to:

• 21

• - 21

• 23

• - 23

If P is a point such that the ratio of the square of the lengths of the tangents from P to the circles x² + y² + 2x - 4y - 20 = 0 and x² + y² - 4x + 2y - 2y - 44 = 0 is 2 : 3, then the locus of P is a circle with centre :

• (7, - 8)

• (- 7, 8)

• (7, 8)

• (- 7, - 8)

If 5x - 12y + 10 = 0 and 12y - 5x + 16 = 0 are two tangents to a circle, then the radius of the circle is

• 1

• 2

• 4

• 6

The eccentricity of the ellipse 9x² + 5y² - 18x - 20y - 16 = 0, is:

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{2}$

• $2$