Mathematics - Conic Sections

Question - 1

The equation of parabola having vertex (0, 0) passing through (2, 3) and axis is X - axis, is

  • A \({ x }^{ 2 }=\frac { 9 }{ 2 } y\)
  • B \({ y }^{ 2 }=\frac { 9 }{ 2 } x\)
  • C \({ y }^{ 2 }=-\frac { 9 }{ 2 } x\)
  • D \({ x }^{ 2 }=-\frac { 9 }{ 2 } y\)

Question - 2

If the points (0, 4) and (0, 2) are respectively the vertex and focus of a parabola, then the equation of the parabola is

  • A \({ x }^{ 2 }+8y=16\)
  • B \({ x }^{ 2 }+8y=32\)
  • C \({ x }^{ 2 }+16y=30\)
  • D \({ x }^{ 2 }-8y=32\)

Question - 3

The equation of the parabola, whose axis is parallel to Y - axis and which passes through the points (0, 4), (1, 9) and (-2, 6), is

  • A \(y=2\left( x+\frac { 3 }{ 4 } \right) ^{ 2 }+\frac { 23 }{ 8 } \)
  • B \(y=2\left( x+\frac { 3 }{ 2 } \right) ^{ 2 }-\frac { 1 }{ 2 } \)
  • C \(y=2\left( x+\frac { 3 }{ 5 } \right) ^{ 2 }+\frac { 1 }{ 2 } \)
  • D \(y=2\left( x+\frac { 1 }{ 2 } \right) ^{ 2 }-\frac { 1 }{ 2 } \)

Question - 4

The equation(s) of the common tangent(s) to the parabolas y2 - 4x - 2y + 5 = 0 and y2 = - 4x is/are

  • A \(x+y=2\\ x-y=1\)
  • B \(x+y=3\\ x-y=2\)
  • C \(x+y=1\\ 4x-2y-1=0\)
  • D \(x+2y=1\\ x+y=-3\)

Question - 5

The locus of the middle points of the chords of the parabola y2 = 4x which touch the parabola x2 = -8y is

  • A \({ y }^{ 3 }-2xy-8=0\)
  • B \({ y }^{ 3 }+2xy-8=0\)
  • C \({ y }^{ 3 }-2xy+8=0\)
  • D \({ y }^{ 3 }-xy-8=0\)

Question - 6

If e is the eccentricity of the ellipse \( \frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) then

  • A \({ b }^{ 2 }={ a }^{ 2 }(1-{ e }^{ 2 })\)
  • B \({ a }^{ 2 }={ b }^{ 2 }(1-{ e }^{ 2 })\)
  • C \({ a }^{ 2 }={ b }^{ 2 }({ e }^{ 2 }-1)\)
  • D \({ b }^{ 2 }={ a }^{ 2 }({ e }^{ 2 }-1)\)

Question - 7

If P is a point on the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 25 } =1\) whose foci are S and S', then PS + PS' is equal to

  • A 8
  • B 7
  • C 5
  • D 10

Question - 8

The equation of the ellipse whose focus is (1, -1), the directrix x - y - 3 = 0 and eccentricity \(\frac { 1 }{ 2 } \) is

  • A \({ 7x }^{ 2 }+2xy+{ 7y }^{ 2 }-10x+10y+7=0\)
  • B \({ 7x }^{ 2 }+2xy+{ 7y }^{ 2 }+7=0\)
  • C \({ 7x }^{ 2 }+2xy+{ 7y }^{ 2 }+10x-10y-7=0\)
  • D None of the above

Question - 9

The number of points outside outside the ellipse on units major axis from which a normal (other than the X - axis) can be drawn to the ellipse is

  • A 0
  • B 3
  • C 5
  • D None of these

Question - 10

From the point A(4, 3), tangents are drawn to the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } =1\) to touch the ellipse at B and C. EF is a tangent to the ellipse parallel to the line BC and toward the point A. The distance of A from EF is equal to.

  • A \(\frac { 4\sqrt { 11 } -5 }{ 2 } \)
  • B \(\frac { 24-4\sqrt { 18 } }{ 5 } \)
  • C \(\frac { 24+4\sqrt { 18 } }{ 5 } \)
  • D \(\frac { 24+2\sqrt { 11 } }{ 5 } \)
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