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

• $\frac{1}{4}$

93.

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

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

Let P be the mid-point of a chord joining the vertex of the parabola y² = 8x to another point on it. Then, the locus of P is

• y² = 2x

• y² = 4x

• $\frac{{\mathrm{x}}^2}{4} + {\mathrm{y}}^2 = 1$

• ${\mathrm{x}}^2 +\hspace{0.17em}\frac{{\mathrm{y}}^2}{4}$ = 1

The line x = 2y intersects the ellipse $\frac{{\mathrm{x}}^2}{4} + {\mathrm{y}}^2 = 1$ at the points P and Q. The equation of the circle with PQ as diameter is

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

• x² + y² = 1

• x² + y² = 2

• x² + y² = $\frac{5}{2}$

The eccentric angle in the first quadrant of a point on the ellipse $\frac{{\mathrm{x}}^2}{10} + \frac{{\mathrm{y}}^2}{8} = 1$  at a distance units from the centre of the ellipse is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$