Mathematics : Conic Section

A line passing through the point of intersection of x + y = 4 and x - y = 2 makes an angle ${\tan }^{-1}\left(\frac{3}{4}\right)$ with the x-axis. It intersects the parabola y² = 4(x-3) at points $\left({\mathrm{x}}_1 , {\mathrm{y}}_1\right) \mathrm{and} \left({\mathrm{x}}_2, {\mathrm{y}}_2\right)$ respectively. Then, $\left|{\mathrm{x}}_1 - {\mathrm{x}}_2\right|$

• $\frac{16}{9}$

• $\frac{32}{9}$

• $\frac{40}{9}$

• $\frac{80}{9}$

The equation of auxiliary circle of the ellipse 16x² + 25y² + 32x - 100y = 284 is

• x² + y² + 2x - 4y - 20 = 0

• x² + y² + 2x - 4y = 0

• (x + 1)² + (y - 2)² = 400

• (x + 1)² + (y - 2)² = 225

If PQ is a double ordinate of the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2} = 1 \mathrm{such} \mathrm{that} ∆\mathrm{OPQ}$ is equilateral. O being the centre. Then, the eccentricity e satisfies 

• $1 < \mathrm{e} < \frac{2}{\sqrt{3}}$

• e = $\frac{2}{\sqrt{2}}$

• e = $\frac{\sqrt{3}}{2}$

• $\mathrm{e} > \frac{2}{\sqrt{3}}$

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