Mathematics · Class 12 Science
Ch 13Parabola — Class 12 Mathematics, concept-first.
The previous two chapters studied a single circle, and then several circles together (system of circles). This chapter turns to a different curve entirely: the parabola. Its name comes from the Greek mathematician Apollonius of Perga (c.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Parabola
A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix) — the conic with eccentricity exactly 1. Choosing axes cleverly turns this locus condition into the familiar equ…
Most relevant Q&A
- Find the coordinates of the points on the parabola $y^2=8x$ whose focal distance is 10.Preview
- Define parabola and obtain the standard form of the parabola $y^2=4ax, \ (a>0)$.Preview
- Find the value of k if the line $2y = 5x + k$ is a tangent to the parabola $y^2 = 6x$.Preview
- Prove that the area of the triangle inscribed in the parabola $y^2 = 4ax$ is $\dfrac{1}{8a} |(y_1 - y_2)(y_2 - y_3)(y_3 - y_1)|$ sq. units w…Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
The previous two chapters studied a single circle, and then several circles together (system of circles). This chapter turns to a different curve entirely: the parabola.
Conic Sections and the Locus Definition of a Parabola
Slice a double cone with a plane and, depending on the angle of the cut, you get a circle, an ellipse, a parabola, or a hyperbola — the four conic sections.
Deriving the Standard Equation y² = 4ax
The locus definition from the previous section gives an equation for the parabola no matter how the focus and directrix happen to sit in the plane, but that equation looks messy unless the axes are ch…
The Four Standard Orientations and the Latus Rectum
The derivation above assumed the parabola opens to the right, with the focus to the right of the vertex.
Parametric Equations and the S-Notation
Every point on can be written in a single-parameter form that is often far more convenient than working with and separately.
Condition for a Tangent and the Equation of the Tangent
A line and a parabola can meet in two points, touch at exactly one point (a tangent), or miss the curve entirely, and all three cases fall straight out of solving them simultaneously.
Equation of the Normal and the Number of Normals
The normal to the parabola at a point is the line through that point perpendicular to the tangent there, and once the tangent's slope is known, the normal's equation is immediate.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Find the coordinates of the points on the parabola $y^2=8x$ whose focal distance is 10.Preview
- Q2Define parabola and obtain the standard form of the parabola $y^2=4ax, \ (a>0)$.Preview
- Q3Find the value of k if the line $2y = 5x + k$ is a tangent to the parabola $y^2 = 6x$.Preview
- Q4Prove that the area of the triangle inscribed in the parabola $y^2 = 4ax$ is $\dfrac{1}{8a} |(y_1 - y_2)(y_2 - y_3)(y_3 - y_1)|$ sq. units w…Preview
- Q5Find the length of the latus rectum of the parabola $y^2=8x$.Preview
- Q6Find the equation of the normal to the parabola $y^2=4ax$ at the point $(at^2,\,2at)$.Preview