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What is a parabola?
What is a parabola?
A parabola is the locus of points that move in a plane such that their distance from a fixed point (focus) is equal to their distance from a fixed line (directrix).
The fixed point is called the ______.
The fixed point is called the ______.
focus
The fixed straight line is called the ______.
The fixed straight line is called the ______.
directrix
Which of the following equations represents a standard parabola with a vertical axis?
Which of the following equations represents a standard parabola with a vertical axis?
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What is the length of the latus rectum for the parabola y^2 = 4ax?
What is the length of the latus rectum for the parabola y^2 = 4ax?
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Find the coordinates of the focus of the parabola y^2 = -12x.
Find the coordinates of the focus of the parabola y^2 = -12x.
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What is the vertex of the parabola x^2 = 6y?
What is the vertex of the parabola x^2 = 6y?
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A double ordinate of the parabola y^2 = 4ax is equal to 8a.
A double ordinate of the parabola y^2 = 4ax is equal to 8a.
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In a double ordinate of the parabola y^2 = 4ax, prove that the lines from the vertex to its two ends are at right angles.
In a double ordinate of the parabola y^2 = 4ax, prove that the lines from the vertex to its two ends are at right angles.
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Study Notes
Definition & Standard Equations
- A parabola is the locus of points where the distance from a fixed point (focus) is equal to the distance from a fixed line (directrix).
- The line perpendicular to the directrix passing through the focus is called the axis of symmetry.
- The intersection point of the parabola and its axis is called the vertex.
- A chord perpendicular to the axis is called a double ordinate. The double ordinate passing through the focus is called the latus rectum.
Standard Parabolas
- For parabolas with vertex at (0, 0) and a > 0:
- y² = 4ax: opens to the right, focus (a, 0), directrix x = -a, length of latus rectum is 4a
- y² = -4ax: opens to the left, focus (-a, 0), directrix x = a, length of latus rectum is 4a
- x² = 4ay: opens upwards, focus (0, a), directrix y = -a, length of latus rectum is 4a
- x² = -4ay: opens downwards, focus (0, -a), directrix y = a, length of latus rectum is 4a
Example 1: y² = -12x
- Focus: (-3, 0)
- Vertex: (0, 0)
- Directrix: x = 3
- Axis of symmetry: y = 0
- Length of latus rectum: 12
Example 2: x² = 6y
- Focus: (0, 3/2)
- Vertex: (0, 0)
- Directrix: y = - 3/2
- Axis of symmetry: x = 0
- Length of latus rectum: 6
Example 3: y² = 4ax with double ordinate of length 8a
- The ends of the double ordinate are (x₁, 4a) and (x₁, -4a)
- Since these points lie on the parabola, they satisfy the equation y² = 4ax.
- Using this, we find x₁ = 4a.
- The slope of the line connecting the vertex to one end of the double ordinate is 4a/4a = 1.
- The slope of the line connecting the vertex to the other end is -4a/4a = -1.
- The product of these slopes is -1, which means the lines are perpendicular.
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Description
This quiz covers the definition, standard equations, and properties of parabolas. It includes concepts such as focus, directrix, axis of symmetry, and latus rectum. Test your understanding of parabolas by answering questions based on these fundamental principles.
