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91.
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

$\frac{9}{2}$

$\frac{9}{2}$

Given equation of parabola is,

y = x²- 4x + 5

$\Rightarrow y = {(x - 2)}^{2} + 1 \Rightarrow {(x - 2)}^{2} = (y - 1) . . . (i)$

and equation of line is,

$y = x + 1 \Rightarrow x - y = - 1 . . . (ii)$

On putting the value of (y - 1) from Eqs. (ii) in (i), we get

${(x - 2)}^{2} = x \Rightarrow x^{2} + 4 - 4 x = x \Rightarrow x^{2} - 5 x + 4 = 0 \Rightarrow x (x - 4) - 1 (x - 4) = 0 \Rightarrow (x - 4) (x - 1) = 0 \Rightarrow x = 1 or x = 4 then from is (ii) y = 2, 5$

$∴ Required area = \int_{1}^{4} \{(x + 1) - (x^{2} - 4 x + 5)\} d x = \int_{1}^{4} (- x^{2} + 5 x - 4) d x = {[\frac{- x^{3}}{3} + \frac{5 x^{2}}{2} - 4 x]}_{1}^{4} = [- \frac{64}{3} + 40 - 16 + \frac{1}{3} - \frac{5}{2} + 4] = (- 21 - \frac{5}{2} + 28) = \frac{9}{2}$

92.

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, $\frac{\tan (∠ PQF)}{\tan (∠ PFQ)}$ 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

94.

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

95.

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

$2 or - \frac{3}{2}$
$- 2 or - \frac{3}{2}$
$2 or \frac{3}{2}$
$- 2 or \frac{3}{2}$

96.

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

97.

Let the foci of the ellipse $\frac{x^{2}}{9} + y^{2} = 1$ 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

98.

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

99.

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

$x^{2} + y^{2} = \frac{1}{2}$
x² + y² = 1
x² + y² = 2
x² + y² = $\frac{5}{2}$

100.

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