The line y = x intersects the hyperbola $\frac{{\mathrm{x}}^2}{9} - \frac{{\mathrm{y}}^2}{25} = 1$ at the points P and Q. The eccentricity of ellipse with PQ as major axis and minor axis of length $\frac{5}{\sqrt{2}}$ is

• $\frac{\sqrt{5}}{3}$

• $\frac{5}{\sqrt{3}}$

• $\frac{5}{9}$

• $\frac{2\sqrt{2}}{9}$

If the distance between the foci of an ellipse is equal to the length of the latusrectum, then its eccentricity is

• $\frac{1}{4}\left(\sqrt{5} - 1\right)$

• $\frac{1}{2}\left(\sqrt{5} + 1\right)$

• $\frac{1}{2}\left(\sqrt{5} - 1\right)$

• $\frac{1}{4}\left(\sqrt{5} + 1\right)$

The equation of the circle passing through the point (1, 1) and the points of intersection of x² + y² -  6x - 8 = 0 and x² + y² - 6 = 0 is 

• x² + y² + 3x - 5 = 0

• x² + y² - 4x + 2 = 0

• x² + y² + 6x - 4 = 0

• x² + y² - 4y - 2 = 0

The area of the region bounded by the parabola y = x² - 4x + 5 and the straight line y= x + l is

• $\frac{1}{2}$

• 2

• 3

• $\frac{9}{2}$

If P be a point on the parabola y = 4ax with focus F. Let Q denote the foot of the perpendicular from P onto the directrix. Then, $\frac{\tan \left(\angle \mathrm{PQF}\right)}{\tan \left(\angle \mathrm{PFQ}\right)}$ is

• 1

• $\frac{1}{2}$

• 2

• $\frac{1}{4}$

The equations of the circles, which touch both the axes and the line 4x + 3y = 12 and have centres in the first quadrant, are

• x² + y² + x - y + 1 = 0

• x² + y² - 2x - 2y + 1 = 0

• x² + y² - 12x - 12y + 36 = 0

• x² + y² - 6x - 6y + 36 = 0

The equation y² + 4x + 4y + k = 0 represents a parabola whose latusrectum is

• 1

• 2

• 3

• 4

If the circles x² + y² + 2x + 2ky + 6 = 0 and x² + y² + 2ky + k = 0 intersect orthogonally, then k is equal to

• $2 \mathrm{or} - \frac{3}{2}$

• $- 2 \mathrm{or} - \frac{3}{2}$

• $2 \mathrm{or} \frac{3}{2}$

• $- 2 \mathrm{or} \frac{3}{2}$

If four distinct points (2k, 3k), (2, 0), (0, 3), (0, 0) lie on a circle, then

• k < 0

• 0 < k < 1

• k = 1

• k > 1

290.

Let the foci of the ellipse $\frac{{\mathrm{x}}^2}{9} + {\mathrm{y}}^2 = 1$ subtend a right angle at a point P. Then, the locus of P is

• x² + y² = 1

• x² + y² = 2

• x² + y² = 4

• x² + y² = 8