Graph of Parabola
Graph of the parabola is a U-shaped curve, which can open either in an upward direction or in a downward direction. Generally, the equation of a parabola which is graphed is written in the form of y = ax2 + bx + c, where a, b, and c are constants that define the shape of the parabola.
If a > 0, in the above equation, the parabola opens in an upward direction and its vertex is the lowest point of the parabola, and if a<0, then the parabola opens in a downward direction and its vertex is the highest point in the parabola. Vertex of the parabola is also the point from where the only line of symmetry of the parabola passes.
Position of Point Relative to the Parabola
Position of a point A (x1, y1) relative to the parabola y2 = 4ax, can be shown using the S1 = y2 – 4ax,
Case 1: If S1 = 0, for any point A, then Point A lies on the parabola.
Case 2: If S1 < 0, for any point A, then Point A lies inside the parabola.
Case 3: If S1 > 0, for any point A, then Point A lies outside the parabola.
Intersection with Straight Line
For a parabola y2 = 4ax, any straight line y = mx + c, can almost intersect the parabola at two points. For the intersection of the line and parabola, put y = mx + c in the equation of the parabola,
(mx + c)2 = 4ax
⇒ m2x2 + c2 + 2mxc = 4ax
⇒ m2x2 + (2mc – 4a)x + c2= 0
⇒ Discriminant = (2mc – 4a)2 – 4m2c2
Case 1: Discriminant > 0:
For positive discriminant, quadratic equations have two real solution,
⇒ There are two point of intersection between line and parabola.
Case 2: Discriminant = 0:
For discriminant equal to 0, quadratic equations have only real solution (common root),
⇒ There are open point of intersection between line and parabola i.e., line is tangent to parabola.
Case 3: Discriminant < 0:
For negative discriminant, quadratic equations have no real roots,
⇒ There are no point of intersection between line and parabola.
Parabola – Graph, Properties, Examples & Equation of Parabola
Parabola is one of the conic sections in Math. It is an intersection of a surface plane and a double-napped cone. A parabola is a U-shaped curve that can be either concave up or down, depending on the equation. Parabolic curves are widely used in many fields such as physics, engineering, finance, and computer sciences.
In this article, we will understand what is a Parabola, its graph, Parabola properties, Parabola examples, and equation of parabola in detail below.
Table of Content
- What is Parabola in Maths?
- Parabola Definition
- Parabola Shape
- Parabola Equation
- Properties of Parabola
- Standard Equation of Parabola
- Important Terms Related to Parabola
- Derivation of Parabola Equation
- Graph of Parabola
- Position of Point Relative to the Parabola
- Intersection with Straight Line
- General Equations of Parabola
- Parametric Coordinates of a Parabola
- Equation of Tangent to a Parabola
- Equation of Tangent in Point Form
- Equation of Tangent in Parametric Form
- Equation of Tangent in Slope Form
- Pair of Tangent from an External Point
- Director Circle of Parabola
- Chord of Contact
- Equation of Normal to a Parabola
- Equation of Normal in Slope Form
- Equation of Normal in Point Form
- Equation of Normal in Parametric Form
- Parabola Formulas
- Parabola Solved Examples
- Practice Questions on Parabola