Introduction
A parabola is a conic section that can be defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas have a wide range of applications in physics, engineering, and other fields. Understanding parabolas is crucial for solving various problems in the JEE Main Mathematics syllabus.
Standard Equation of a Parabola
The standard form of a parabola's equation depends on its orientation. There are two primary orientations:
Parabola Opening Upward or Downward
The standard form is: $$ y^2 = 4ax $$
- Vertex: The vertex of the parabola is at the origin (0, 0).
- Focus: The focus is at (a, 0).
- Directrix: The directrix is the line $x = -a$.
- Axis of Symmetry: The axis of symmetry is the x-axis.
For the equation ( y^2 = 12x ):
- Here, ( 4a = 12 \implies a = 3 ).
- Focus: (3, 0)
- Directrix: ( x = -3 )
Parabola Opening Rightward or Leftward
The standard form is: $$ x^2 = 4ay $$
- Vertex: The vertex of the parabola is at the origin (0, 0).
- Focus: The focus is at (0, a).
- Directrix: The directrix is the line $y = -a$.
- Axis of Symmetry: The axis of symmetry is the y-axis.
For the equation ( x^2 = -8y ):
- Here, ( 4a = -8 \implies a = -2 ).
- Focus: (0, -2)
- Directrix: ( y = 2 )
General Equation of a Parabola
A parabola can also be represented in its general form: $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$
However, for JEE Main, we often deal with the specific cases where either $A$ or $C$ is zero, simplifying the analysis.
Vertex Form of a Parabola
The vertex form of a parabola's equation is useful for identifying its vertex directly: $$ y = a(x - h)^2 + k $$ where $(h, k)$ is the vertex of the parabola.
ExampleFor the equation ( y = 2(x - 1)^2 + 3 ):
- Vertex: (1, 3)
- Opens upward since ( a = 2 > 0 ).
Parametric Form
The parametric equations of a parabola ( y^2 = 4ax ) are given by: $$ x = at^2 $$ $$ y = 2at $$
where ( t ) is a parameter.
NoteParametric equations are particularly useful in problems involving motion along a parabolic path.
Focal Distance
The distance of any point ( (x, y) ) on the parabola ( y^2 = 4ax ) from the focus ( (a, 0) ) is: $$ \sqrt{(x - a)^2 + y^2} $$
Latus Rectum
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry and passing through the focus. Its length is given by ( 4a ).
For ( y^2 = 4ax ):
- Length of latus rectum: ( 4a )
- Endpoints: ( (a, 2a) ) and ( (a, -2a) )
Tangents and Normals
Tangent to a Parabola
For the parabola ( y^2 = 4ax ), the equation of the tangent at point ( (x_1, y_1) ) is: $$ yy_1 = 2a(x + x_1) $$
ExampleFind the equation of the tangent to the parabola ( y^2 = 4x ) at the point ( (1, 2) ):
- Using the formula, ( y \cdot 2 = 2 \cdot 1 \cdot (x + 1) )
- Simplifying, ( 2y = 2(x + 1) \implies y = x + 1 )
Normal to a Parabola
The equation of the normal to the parabola ( y^2 = 4ax ) at point ( (x_1, y_1) ) is: $$ y - y_1 = -\frac{y_1}{2a}(x - x_1) $$
ExampleFind the equation of the normal to the parabola ( y^2 = 4x ) at the point ( (1, 2) ):
- Using the formula, ( y - 2 = -\frac{2}{2}(x - 1) )
- Simplifying, ( y - 2 = -x + 1 \implies y = -x + 3 )
Reflection Property
One of the key properties of a parabola is its reflection property: any ray parallel to the axis of symmetry and directed towards the parabola is reflected through the focus.
TipThis property is crucial in designing parabolic reflectors and antennas.
Common Mistakes
Common MistakeConfusing the equations of parabolas opening upwards/downwards with those opening rightwards/leftwards.
Common MistakeForgetting to square the parameter ( t ) in parametric equations.
Summary
Understanding the properties and equations of parabolas is essential for solving various problems in the JEE Main Mathematics syllabus. Key concepts include the standard forms, vertex form, parametric form, tangents, normals, and the reflection property. Mastery of these topics will greatly aid in tackling related questions efficiently.
