The locus of the foot of perpendicular drawn from the centre of

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1.

The locus of the foot of perpendicular drawn from the centre of the ellipse x2+3y2 =6 on any tangent to it is

  • (x2-y2)2 = 6x2+2y2

  • (x2-y2)2 = 6x2 -2y2

  • (x2+y2)2 = 6x2+2y2

  • (x2+y2)2 = 6x2+2y2


C.

(x2+y2)2 = 6x2+2y2

Equation of ellipse is,

straight x squared space plus 3 straight y squared space equals space 6 space or space straight x squared over 6 space plus straight y squared over 2 space equals space 1
Equaion space of space the space tangent space is space
fraction numerator straight x space cos space straight theta over denominator straight a end fraction plus fraction numerator straight y space sin space straight theta over denominator straight b end fraction space equals space 1
Let space left parenthesis straight h comma straight k right parenthesis space be space any space point space of space the space locus
therefore space straight h over straight a cos space straight theta space plus straight k over straight b sinθ space equals space 1 space... space left parenthesis straight i right parenthesis
Slope space of space the space tangent space line space is space fraction numerator negative straight b over denominator straight a end fraction cot space straight theta.
The slope of the perpendicular drawn from the centre (0,0) to (h,k) is k/h.
Since, both the lines are perpendicular,
therefore space open parentheses straight k over straight h close parentheses space straight x open parentheses negative straight b over straight a space cot space straight theta close parentheses space equals space minus space 1
rightwards double arrow space cos space straight theta space equals space αha comma space sin space straight theta space equals space αkb
From space eq space left parenthesis straight i right parenthesis space we space get
straight h over straight a left parenthesis αha right parenthesis space plus space straight k over straight b space left parenthesis αkb right parenthesis space equals space 1
rightwards double arrow space straight h squared straight alpha space plus straight k squared straight alpha space equals space 1
rightwards double arrow straight alpha space equals space fraction numerator 1 over denominator straight h squared plus straight k squared end fraction
Also comma space sin squared straight theta space plus space Cos squared space straight theta space equals 1
rightwards double arrow space left parenthesis αkb right parenthesis squared space plus space left parenthesis αha right parenthesis squared space equals 1
rightwards double arrow space straight alpha squared straight k squared straight b squared space plus straight alpha squared straight h squared straight a squared space equals 1
rightwards double arrow fraction numerator straight k squared straight b squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction plus fraction numerator straight h squared straight a squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction space equals 1
fraction numerator 2 straight k squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction plus fraction numerator 6 straight h squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction space equals 1
6 straight x squared plus 2 straight y squared space equals space left parenthesis straight x squared plus straight y squared right parenthesis squared
 

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2.

The slope of the line touching both the parabolas y2 = 4x and x2-32y is 

  • 1/2

  • 3/2

  • 1/8

  • 1/8

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3.

The circle passing through (1,-2) and touching the axis of x at (3,0) also passes through the point

  • (-5,2)

  • (2,-5)

  • (5,-2)

  • (5,-2)

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4.

The equation of the circle passing through the foci of the ellipse straight x squared over 16 space plus straight y squared over 9 space equals 1 and having centre at (0,3) is 

  • x2+y2-6y-7 =0

  • x2+y2-6y+7 =0

  • x2+y2-6y-5 =0

  • x2+y2-6y-5 =0

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5.

Let Tn be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1 − Tn = 10, then the value of n is

  • 7

  • 5

  • 10

  • 10

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6.

Given A circle, 2x2 + 2y2= 5 and parabola, straight y squared space equals space 4 square root of 5 space straight x 
Statement I An equation of a common tangent to these curves is straight y space equals straight x plus square root of 5
Statement II If the line straight y space equals space mx space plus fraction numerator space square root of 5 over denominator straight m end fraction space left parenthesis straight m space not equal to 0 right parenthesis is the common tangent, then m satisfies m4-3m2+2 =0

  • Statement I is true,  Statement II is true; Statement II is a correct explanation for statement I

  • Statement I is true, Statement II is true; Statement II is not a correct explanation for statement I

  • Statement I is true, Statement II is false

  • Statement I is true, Statement II is false

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7.

Statement I An equation of a common tangent to the parabola straight y squared space equals space 16 space square root of 3 straight x end root and the ellipse 2x2 +y2 =4 is space straight y space equals 2 straight x space plus 2 square root of 3.
Statement II If the line straight Y space equals space mx space plus fraction numerator 4 square root of 3 over denominator straight m end fraction comma space left parenthesis straight m space not equal to 0 right parenthesis is a common tangent to the parabola straight y squared space equals space 16 space square root of 3 straight x and the ellipse 2x2 +y2 =4, then m satisfies m4 +2m2 =24

  • Statement 1 is false, statement 2 is true

  • Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

  • Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

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8.

The area bounded between the parabolas x2=y/4 and x2 = 9y, and the straight line y = 2 is

  • 20 square root of 2
  • fraction numerator 10 space square root of 2 over denominator 3 end fraction
  • fraction numerator 20 space square root of 2 over denominator 3 end fraction
  • fraction numerator 20 space square root of 2 over denominator 3 end fraction
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9.

If z ≠ 1 and fraction numerator straight z squared over denominator straight z minus 1 end fraction is real, then the point represented by the complex number z lies

  • either on the real axis or on a circle passing through the origin

  • on a circle with centre at the origin

  • either on the real axis or on a circle not passing through the origin

  • either on the real axis or on a circle not passing through the origin

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10.

The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is

  • 10/3

  • 3/5

  • 6/5

  • 6/5

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