The transverse axis of a hyperbola is along the x - axis and its length is 2a. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is

• 6x² - y² = 3a²

• x² - 3y² = 3a²

• x² - 6y² = 3a²

• 3x² - y² = 3a²

A point moves in such a way that the difference of its distance from two points (8, 0) and (- 8, 0) always remains 4. Then, the locus of the point is

• a circle

• a parabola

• an ellipse

• a hyperbola

Let C₁ and C₂ denote the centres of the circles x² + y² = 4 and (x - 2)² + y² = 1 respectively and let P and Q be their points of intersection. Then, the areas of $∆$C₂PQ and CPQ are in the ratio

• 3 : 1

• 5 : 1

• 7 : 1

• 9 : 1

The incentre of an equilateral triangle is (1, 1) and the equation of one side is 3x + 4y + 3 = 0. Then, the equation of the circumcircle of the triangle is

• x² + y² - 2x - 2y - 2 = 0

• x² + y² - 2x - 2y - 14 = 0

• x² + y² - 2x - 2y + 2 = 0

• x² + y² - 2x - 2y + 14 = 0

The eccentricity of the hyperbola 4x² - 9y² = 36 is

• $\frac{\sqrt{11}}{3}$

• $\frac{\sqrt{15}}{3}$

• $\frac{\sqrt{13}}{3}$

• $\frac{{\displaystyle \sqrt{14}}}{{\displaystyle 3}}$

106.

The length of the latus rectum of the ellipse 16x² + 25y² = 400 is

• 5/16 unit

• 32/5 unit

• 16/5 unit

• 5/32 unit

The vertex of the parabola y² + 6x - 2y + 13 = 0 is

• (1, - 1)

• (- 2, 1)

• $\left(\frac{3}{2}, 1\right)$

• $\left(- \frac{7}{2}, 1\right)$

The coordinates of a moving point P are (2t² + 4, 4t + 6). Then, its locus will be

• circle

• straight line

• parabola

• ellipse

The equation 8x² + 12y² - 4x + 4y - 1 = 0 represents

• an ellipse

• a hyperbola

• a parabola

• a circle

If the straight line y = mx lies outside the circle x² + y² - 20y + 90 = 0, then the value of m will satisfy

• m < 3

• $\left|\mathrm{m}\right| < 3$

• m > 3

• $\left|\mathrm{m}\right| > 3$