The locus of centre of circles which cuts orthogonally the circle

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 Multiple Choice QuestionsMultiple Choice Questions

251.

Equation of tangent to the circle x2 + y2 - 2x - 2y + 1 = 0 perpendicular to y = x is given by

  • x + y ± 1 = 0

  • x + y = 2 ± 3

  • x + y ± 3 = 0

  • None of these


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252.

The locus of centre of circles which cuts orthogonally the circle x2 + y2 - 4x + 8 = 0 and touches x + 1 = 0, is

  • y2 + 6x + 7 = 0

  • x2 + y2 + 2x + 3 = 0

  • x2 + 3y + 4 = 0

  • None of the above


A.

y2 + 6x + 7 = 0

x2 + y2 + 2gx + 2fy + c = 0

Given equation of circle is

x2 + y2 - 4x + 8 = 0

The centres of above circles are (- g, - f) and (2, 0)

Condition of orthogonality is

2(g1g2 + f1f2) = c1 + c2

 2g × - 2 + f × 0 = c+ 8                             - 4g = c + 8       ...i

Also, the assume circle touch the lme x + 1 = 0.

The perpendicular drawn from centre to the line is equal to radius

   - g + 112 = g2 + f2 - c -   g + 1 = g2 + f2 - c

On squaring both sides, we get

g2 + 1 - 2 = g2 + f2 - c            c = f2 + 2g - 1

Putting the value of c in Eq (i), we get

4g = f2 + 2 - 1 + 8

 f2 + 2g + 4g + 7 = 0          f2 + 6g + 7 = 0

Thus, locus of circle is y2 + 6x + 7 = 0.


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253.

The condition for the line lx + my + n = 0 to be a normal to x225 + y29 = 1 is

  • l29 + m2l25 = n2256

  • 9m2 + 25l2 = 256n2

  • l29 - m2l25 = n2256

  • None of these


254.

The radical centre of the system of circles

            x2 + y2 + 4x + 7 = 0,

2(x2 + y2) + 3x + 5y + 9 = 0

and               x2 + y2 + y = 0 is

  • (- 2, - 1)

  • (1, - 2)

  • (- 1, - 2)

  • None of these


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255.

The point on the straight line y = 2x + 11 which is nearest to the circle 16(x2 + y) + 32x - 8y - 50 = 0, is

  • 92, 2

  • 92, - 2

  • - 92, 2

  • - 92, - 2


256.

The locus of the extrimities of the latusrectum of the family of ellipses b2x2 + y2 = a2b2 having a given major axis, is

  • x2 ± ay = a2

  • y2 ± bx = a2

  • x2 ± by = a2

  • y2 ± ax = b2


257.

The number of common tangents to two circles x2 + y2 = 4 and x2 + y2 - 8x + 12 = 0 is

  • 1

  • 2

  • 3

  • 4


258.

If the tangent at the point 2secθ, 3tanθ of the hyperbola x24 - y29 = 1 is parallel to 3x - y + 4 = 0, then the value of θ is

  • π4

  • π3

  • π6

  • π2


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259.

If an equilateral triangle is inscribed in the circle x2 + y2 = a2, the lenth of its each side is

  • 2a

  • 3a

  • 32a

  • 13a


260.

If the vertex is (3,0) and the extremities of the latusrectum are (4, 3) and (4, - 3), then the equation of the parabola is

  • y2 = 4(x - 3)

  • x2 = 4(y - 3)

  • y2 = - 4(x + 3)

  • x2 = - 4(y + 3)


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