If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

• 2 < a < 4

• $- \frac{1}{2} < \mathrm{a} < 2$

• 0 < a < 2

• $- \frac{1}{2} < \mathrm{a} < \frac{3}{2}$

The locus of the mid-points of chords of the circle x² + y² = 1, which subtends a right angle at the origin, is

• x² + y² = $\frac{1}{4}$

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

• xy = 0

• x² - y² = 0

• x = - a

• x = a

• x = 0

• x = - $\frac{\mathrm{a}}{2}$

The points of the ellipse 16x² + 9y² = 400 at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

• $\left(3, \frac{16}{3}\right) \mathrm{and} \left(- 3, - \frac{16}{3}\right)$

• $\left(3, - \frac{16}{3}\right) \mathrm{and} \left(- 3, \frac{16}{3}\right)$

• $\left(\frac{1}{16}, \frac{1}{9}\right) \mathrm{and} \left(- \frac{1}{16}, - \frac{1}{9}\right)$

• $\left(\frac{1}{16}, - \frac{1}{9}\right) \mathrm{and} \left(- \frac{1}{16}, \frac{1}{9}\right)$

If the parabola x² = ay makes an intercept of length $\sqrt{40}$ units on the line y - 2x = 1, then a is equal to

• 1

• - 2

• - 1

• 2

If the vertex of the conic y - 4y = 4x - 4a always lies between the straight lines x + y = 3 and 2x + 2y - 1 = 0, then

• 2 < a < 4

• $- \frac{1}{2} < \mathrm{a} < 2$

• 0 < a < 2

• $- \frac{1}{2} < \mathrm{a} < \frac{3}{2}$

Number of intersecting points of the conics 4x² + 9y² = 1 and 4x² + y² = 4 is

• 1

• 2

• 3

• 0

68.

If the point $\left(2\cos \left(\mathrm{\theta }\right), 2\sin \left(\mathrm{\theta }\right)\right) \mathrm{for} (0, 2\mathrm{\pi })$ lies in the region between the lines x + y = 2 and x - y = 2 containing the origin, then 0 lies in

• $\left(0, \frac{\mathrm{\pi }}{2}\right) \cup \left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$

• $\left[0, \mathrm{\pi }\right]$

• $\left(\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right)$

• $\left[\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right]$

Let 16x² - 3y - 32x - 12y = 44 represents a hyperbola. Then,

• length of the transverse axis is $2\sqrt{3}$

• length of each latusrectum is $32/\sqrt{3}$

• eccentricity is $\sqrt{19/3}$

• equation of a directrix is x = $\frac{\sqrt{19}}{3}$

If the straight line (a - 1)x - by + 4 = 0 is normal to the hyperbola xy = 1, then which of the following does not hold?

• a > 1, b > 0

• a > 1, b < 0

• a < 1, b < 0

• a < 1, b > 0