The area included between the parabola y = x24a&nb

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 Multiple Choice QuestionsMultiple Choice Questions

401.

The distance between the focii of the ellipse
x = 3cosθ, y = 4sinθ is

  • 27

  • 72

  • 7

  • 37


402.

The equations of the latus rectum of the ellipse
9x2 + 25y2 - 36x + 50y - 164 = 0 are

  • x - 4 = 0, x + 2 = 0

  • x - 6 = 0, x + 2 = 0

  • x + 6 = 0, x - 2 = 0

  • x + 4 = 0, x + 5 = 0


403.

The values of m for which the line y = mx + 2
becomes a tangent to the hyperbola 4x2 - 9y2 = 36 is

  • ± 23

  • ± 223

  • ± 89

  • ± 423


404.

The equation of the common tangent drawn to the curves y = 8x and xy = - 1 is

  • y = 2x + 1

  • 2y = x + 6

  • y = x + 2

  • 3y = 8x + 2


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405.

The area included between the parabola y = x24a and the curve y = 8a3x2 + 4a2 is

  • a22π + 23

  • a22π - 83

  • a2π + 43

  • a22π - 43


D.

a22π - 43

For point of intersection, equate both equationsy = x24a and the curve y = 8a3x2 + 4a2 x2x2 + 4a2 = 4a8a3 x4 + 4a2x2 = 32a4 x4 + 4a2x2 - 32a4 = 0 x2x2 + 8a2 - 4a2x2 + 8a2 = 0 x2 + 8a2x2 - 4a2 = 0 x2 - 4a2 = 0 or x2 + 8a2 = 0 x2 = 4a2 or x2 = - 8a2 x2 = 4a2

  x = ± 2a  x = 2aHence, we take limits from 0 to 2a. Area between 2 graphs   = 2 × 02a8a3x2 + 4a2 - x24adx   = 2 × 02a8a3x2 + 4a2dx - 02ax24adx   = 2 × 8a3 × 02a1x2 + 4a2dx - 14a02ax2dx   = 2 × 8a312atan-1x2a02a - 14ax3302a   = 2 × 8a32atan-12a2a - tan-102a - 14a2a33 - 033

   = 2 × 4a2tan-11 - tan-10 - 14a8a33 - 0   = 2 × 4a2π4 - 0 - 14a8a33   = 2 × 4a2 × π4 - 14a × 8a33   = 2 × a2π - 2a23 = 2a2π - 23  = 2a2π - 23   = a22π - 43


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406.

If a circle with radius 2.5 units passes through the points (2, 3) and (5, 7), then its centre is

  • (1 5, 2)

  • (7, 10)

  • (3, 4)

  • (3 5, 5)


407.

The circumcentre of the triangle formed by the points (1, 2, 3) (3, - 1, 5), (4, 0, - 3) is

  • (1, 1, 1)

  • (2, 2, 2)

  • (3, 3, 3)

  • 72,  - 12, 1


408.

The lines y = 2x + 76 and 2y + x = 8 touch the ellipse x2 + y= 1. If the point of x216 + y212 = 1 intersection of these two lines lie on a circle, whose centre coincides with the centre of that ellipse, then the equation of that circle is

  • x2 + y2 = 28

  • x2 + y2 = 12

  • x2 + y2 = 12

  • x2 + y2 = 4 + 82


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409.

If lx + my = 1 is a normal to the hyperbola x2a2 - y2b2 = 1, then a2m2 - b2l2 = ?

  • m2l2a2 + b22

  • l2 + m2(a2 + b2)2

  • l2m2a2 + b22

  • l2m2(a2 + b2)2


410.

x - 13x + 4 < x - 33x - 2 holds, for all x in the internal

  •  - 43, 23

  • ,  - 54

  • 33, 

  •  - ,  - 54  34, - 


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