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Notely Maths 12
Class 12 Maths
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6.4 Q-1

Question Statement

Find the focus, vertex, and direction of the parabola for the given equations, and sketch their graphs.

Background and Explanation

This problem focuses on analyzing and graphing parabolas. A parabola is defined as the set of all points equidistant from a point called the focus and a line called the directrix. Parabolas can be oriented in different directions, depending on their equations. The standard forms are:

  1. y2=4axy^2 = 4ax (opens right), y2=−4axy^2 = -4ax (opens left)
  2. x2=4ayx^2 = 4ay (opens upward), x2=−4ayx^2 = -4ay (opens downward)

To solve these problems, we compare the given equations with the standard forms, extract relevant parameters (e.g., aa), determine key features (focus, vertex, directrix), and then sketch the graphs.

Solution

i. y2=8xy^2 = 8x

  • Compare with y2=4axy^2 = 4ax, so:
4a=8  ⟹  a=2 4a = 8 \implies a = 2
  • Focus: (a,0)=(2,0)(a, 0) = (2, 0)
  • Vertex: (0,0)(0, 0)
  • Directrix: x=−a=−2x = -a = -2
  • Axis: Along the xx-axis.

Diagram Placeholder

ii. x2=−16yx^2 = -16y

  • Compare with x2=−4ayx^2 = -4ay, so:
4a=16  ⟹  a=4 4a = 16 \implies a = 4
  • Focus: (0,−4)(0, -4)
  • Vertex: (0,0)(0, 0)
  • Directrix: y=4y = 4
  • Axis: Along the yy-axis.

Diagram Placeholder

iii. x2=5yx^2 = 5y

  • Compare with x2=4ayx^2 = 4ay, so:
4a=5  ⟹  a=54 4a = 5 \implies a = \frac{5}{4}
  • Focus: (0,54)\left( 0, \frac{5}{4} \right)
  • Vertex: (0,0)(0, 0)
  • Directrix: y=−54y = -\frac{5}{4}
  • Axis: Along the yy-axis.

Diagram Placeholder

iv. y2=−12xy^2 = -12x

  • Compare with y2=−4axy^2 = -4ax, so:
4a=12  ⟹  a=3 4a = 12 \implies a = 3
  • Focus: (−3,0)(-3, 0)
  • Vertex: (0,0)(0, 0)
  • Directrix: x=3x = 3
  • Axis: Along the xx-axis.

Diagram Placeholder

v. x2=4(y−1)x^2 = 4(y - 1)

  • Shift origin to (0,1)(0, 1), rewrite as:
x2=4y x^2 = 4y
  • Compare with x2=4ayx^2 = 4ay, so:
a=1 a = 1
  • Focus: (0,2)(0, 2)
  • Vertex: (0,1)(0, 1)
  • Directrix: y=0y = 0
  • Axis: Along the yy-axis.

Diagram Placeholder

vi. y2=−8(x−3)y^2 = -8(x - 3)

  • Shift origin to (3,0)(3, 0), rewrite as:
y2=−8x y^2 = -8x
  • Compare with y2=−4axy^2 = -4ax, so:
4a=8  ⟹  a=2 4a = 8 \implies a = 2
  • Focus: (1,0)(1, 0)
  • Vertex: (3,0)(3, 0)
  • Directrix: x=5x = 5
  • Axis: Along the xx-axis.

Diagram Placeholder

vii. (x−1)2=8(y+2)(x - 1)^2 = 8(y + 2)

  • Shift origin to (1,−2)(1, -2), rewrite as:
x2=8y x^2 = 8y
  • Compare with x2=4ayx^2 = 4ay, so:
4a=8  ⟹  a=2 4a = 8 \implies a = 2
  • Focus: (1,0)(1, 0)
  • Vertex: (1,−2)(1, -2)
  • Directrix: y=−4y = -4
  • Axis: Along the yy-axis.

Diagram Placeholder

viii. y=6x2−1y = 6x^2 - 1

  • Rewrite as:
x2=16(y+1) x^2 = \frac{1}{6}(y + 1)
  • Compare with x2=4ayx^2 = 4ay, so:
4a=16  ⟹  a=124 4a = \frac{1}{6} \implies a = \frac{1}{24}
  • Focus: (0,−2324)\left(0, -\frac{23}{24}\right)
  • Vertex: (0,−1)(0, -1)
  • Directrix: y=−2524y = -\frac{25}{24}
  • Axis: Along the yy-axis.

Diagram Placeholder

ix. x+8−y2+2y=0x + 8 - y^2 + 2y = 0

  • Rewrite as:
(y−1)2=x+9 (y - 1)^2 = x + 9
  • Shift origin to (−9,1)(-9, 1), rewrite as:
y2=x y^2 = x
  • Compare with y2=4axy^2 = 4ax, so:
4a=1  ⟹  a=14 4a = 1 \implies a = \frac{1}{4}
  • Focus: (−354,1)\left( -\frac{35}{4}, 1 \right)
  • Vertex: (−9,1)(-9, 1)
  • Directrix: x=−374x = -\frac{37}{4}
  • Axis: Along the xx-axis.

Diagram Placeholder

x. x2−4x−8y+4=0x^2 - 4x - 8y + 4 = 0

  • Rewrite as:
(x−2)2=8y (x - 2)^2 = 8y
  • Shift origin to (2,0)(2, 0), rewrite as:
x2=8y x^2 = 8y
  • Compare with x2=4ayx^2 = 4ay, so:
4a=8  ⟹  a=2 4a = 8 \implies a = 2
  • Focus: (2,2)(2, 2)
  • Vertex: (2,0)(2, 0)
  • Directrix: y=−2y = -2
  • Axis: Along the yy-axis.

Diagram Placeholder

Key Formulas or Methods Used

  • Standard Form of Parabola:
y2=4axorx2=4ay y^2 = 4ax \quad \text{or} \quad x^2 = 4ay
  • Focus: Determined using aa value.
  • Vertex: The origin or the shifted origin.
  • Directrix: Line opposite the focus, located at x=−ax = -a or y=−ay = -a.

Summary of Steps

  1. Rewrite the equation to match the standard form of a parabola.
  2. Identify aa, focus, vertex, and directrix.
  3. Shift the origin if needed.
  4. Sketch the graph using the identified parameters.