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When a plane intersects one nappe of a cone at any place other than the vertex, such that angle beta between the plane and the axis of the cone is equal to angle alpha between the axis and the generator of the cone, the resulting conic section is a parabola.

If a solid object is thrown in the air at an angle to the ground, the trajectory of the object is a parabola.

Consider a fixed straight line L in a plane. Also consider a fixed point F in the same plane but not on line L. Now consider points P1, P2 and P3 such that their distances from the fixed line, L, and the fixed point, F, are the same.

Then the curve passing through all such points is a parabola. Thus, a parabola is the set of all the points in a plane, that are equidistant from a fixed line and a fixed point in the plane.

The curve of a parabola forms only if the fixed point, F, does not lie on the fixed line, L. If point F lies on line L, then the parabola becomes a straight line. This is also called the degenerate case of a parabola.

The fixed point, F, in the plane is called the focus of the parabola.

The fixed line, L, in the plane is called the directrix of the parabola.

The line passing through the focus and perpendicular to the directrix is called the axis of the parabola.

A parabola is symmetrical about its axis. Thus, the axis of a parabola is also called its axis of symmetry. The point where the axis intersects the curve of a parabola is called the vertex of the parabola.

The line segment passing through the focus, perpendicular to the axis and with its end points on the curve of the parabola is called the latus rectum of the parabola.

**Standard Equations of a Parabola**

Consider a parabola with its vertex at the origin and its axis along the X- or Y-axis, then there are four distinct possibilities.

Case I: The axis of symmetry of the parabola is along the X-axis with the focus of the parabola on its positive side and the directrix cutting the negative side. Since vertex O is also a point on the parabola, its distance from the focus and the directrix will be the same.

Thus, if the axis of symmetry lies along the X-axis, the vertex lies at the origin and the focus at (a, 0), then the directrix will intersect the X-axis at (-a, 0).

Focus (*a*, 0), Directrix *x* = -*a*

Case II: The axis of symmetry is along the X-axis with the focus of the parabola on its negative side and the directrix cutting the positive side.

Focus (-*a*, 0), Directrix *x* = *a*

Case III: The axis of symmetry is along the Y-axis with the focus of the parabola on its positive side, and the directrix cutting the negative side of the Y-axis.

Focus (0, *a*), Directrix *y* = -*a*

Case IV: The axis of symmetry is along the Y-axis, the focus of the parabola on its negative side and the directrix cutting the positive side.

**Equation of the parabola for case I**

Consider a point P on the parabola with the coordinates (x, y).

Join P to F. PF is the distance of point P from the focus of the parabola.

Let PB be the perpendicular drawn from point P on the directrix. Thus, PB is the distance from point P to the directrix.

Since point B lies on the directrix, its x-coordinate will be -a. The y-coordinate of point B will be the same as that of point P, that is, y.

The distance between the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) = √(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2} ^{ }

Substituting the coordinates of points P and F in the distance formula,

PF = √(*x* - *a*)^{2} + (*y* - 0)^{2}

⇒ ^{ }PF = √(*x* - *a*)^{2} + *y*^{2} ……(1)

Similarly, substituting the coordinates of points P and B in the distance formula,

PB = √{*x* - (-*a*)}^{2} + (*y* - *y*)^{2}

⇒ ^{ }PB = √(*x* + *a*)^{2} ……(2)

PF = PB

⇒ √(*x* - *a*)^{2} + *y*^{2} = √(*x* + *a*)^{2}

Squaring on both sides, we get

(*x* - *a*)^{2} + *y*^{2} = (*x* + *a*)^{2}

⇒ *y*^{2} = (*x* + *a*)^{2} - (*x* - *a*)^{2}

⇒ *y*^{2 }*=* x^{2} + 2ax + a^{2}* - (*x^{2} - 2ax + a^{2}*)*

⇒ *y*^{2 }*=* x^{2} + 2ax + a^{2}* - *x^{2} + 2ax - a^{2}* *

⇒*y*^{2} = 4*ax*

The equation y^{2} = 4ax represents the given parabola.

Substituting the value of Y square as 4AX in equation 1,

*y*^{2} = 4*ax*

PF = √(*x* - *a*)^{2} + *y*^{2}

= √(*x* - *a*)^{2} + 4*ax*

= √*x*^{2} - 2*ax* + *a*^{2} + 4*ax*

= √*x*^{2} + 2*ax* + *a*^{2}

= √(*x* + *a*)^{2} = PB

Thus, the equation y^{2} = 4ax represents the parabola with the axis of symmetry along the X-axis, the vertex at the origin, the focus at (a, 0) , and directrix is x = -a.

**Equation of the parabola for case II**

By similar derivation, the equation of the parabola with the axis of symmetry along the X-axis, the vertex at the origin, the focus at (-a, 0), and directrix x= a, is y^{2} = -4ax.

**Equation of the parabola for case III**

The equation of the parabola with the axis of symmetry along the Y-axis, the vertex at the origin, the focus at (a, 0), and directrix y = -a, is *x*^{2} = 4*ay*.

**Equation of the parabola for case IV**

The equation of the parabola with the axis of symmetry along the Y-axis, the vertex at origin, the focus at (0, -a), and directrix y =a is *x*^{2} = -4*ay.*

These four equations are called the standard equations of a parabola.

If the equation of the parabola has the term *y*^{2}, then the axis of symmetry lies along the *X*-axis.

If the equation of the parabola has the term *x*^{2}, then the axis of symmetry lies along the *Y*-axis.

If the axis of symmetry lies along the *X*-axis, then the parabola opens to right if F lies on the positive side of the *X*-axis.

If the axis of symmetry lies along the *X*-axis, then the parabola opens to left if F lies on the negative side of the *X*-axis.

If the axis of symmetry lies along the *Y*-axis, then the parabola opens upwards if F lies on the positive side of the *Y*-axis.

If the axis of symmetry lies along the *Y*-axis, then the parabola opens downwards if F lies on the negative side of the *Y*-axis.

**Length of the Latus Rectum**

Consider a parabola with latus rectum AB.

Let OF = *a*

OM = OF

⇒ FM = 2*a*

FM = AC (distance of point A from directrix)

⇒ AC = 2*a*

By definition, AC = AF

⇒ AF = 2*a*

Since a parabola is symmetrical about its axis,

AF = FB

⇒ FB = 2a

⇒ Length of the Latus Rectum (AB) = AF + FB = 4a

When a plane intersects one nappe of a cone at any place other than the vertex, such that angle beta between the plane and the axis of the cone is equal to angle alpha between the axis and the generator of the cone, the resulting conic section is a parabola.

If a solid object is thrown in the air at an angle to the ground, the trajectory of the object is a parabola.

Consider a fixed straight line L in a plane. Also consider a fixed point F in the same plane but not on line L. Now consider points P1, P2 and P3 such that their distances from the fixed line, L, and the fixed point, F, are the same.

Then the curve passing through all such points is a parabola. Thus, a parabola is the set of all the points in a plane, that are equidistant from a fixed line and a fixed point in the plane.

The curve of a parabola forms only if the fixed point, F, does not lie on the fixed line, L. If point F lies on line L, then the parabola becomes a straight line. This is also called the degenerate case of a parabola.

The fixed point, F, in the plane is called the focus of the parabola.

The fixed line, L, in the plane is called the directrix of the parabola.

The line passing through the focus and perpendicular to the directrix is called the axis of the parabola.

A parabola is symmetrical about its axis. Thus, the axis of a parabola is also called its axis of symmetry. The point where the axis intersects the curve of a parabola is called the vertex of the parabola.

The line segment passing through the focus, perpendicular to the axis and with its end points on the curve of the parabola is called the latus rectum of the parabola.

**Standard Equations of a Parabola**

Consider a parabola with its vertex at the origin and its axis along the X- or Y-axis, then there are four distinct possibilities.

Case I: The axis of symmetry of the parabola is along the X-axis with the focus of the parabola on its positive side and the directrix cutting the negative side. Since vertex O is also a point on the parabola, its distance from the focus and the directrix will be the same.

Thus, if the axis of symmetry lies along the X-axis, the vertex lies at the origin and the focus at (a, 0), then the directrix will intersect the X-axis at (-a, 0).

Focus (*a*, 0), Directrix *x* = -*a*

Case II: The axis of symmetry is along the X-axis with the focus of the parabola on its negative side and the directrix cutting the positive side.

Focus (-*a*, 0), Directrix *x* = *a*

Case III: The axis of symmetry is along the Y-axis with the focus of the parabola on its positive side, and the directrix cutting the negative side of the Y-axis.

Focus (0, *a*), Directrix *y* = -*a*

Case IV: The axis of symmetry is along the Y-axis, the focus of the parabola on its negative side and the directrix cutting the positive side.

**Equation of the parabola for case I**

Consider a point P on the parabola with the coordinates (x, y).

Join P to F. PF is the distance of point P from the focus of the parabola.

Let PB be the perpendicular drawn from point P on the directrix. Thus, PB is the distance from point P to the directrix.

Since point B lies on the directrix, its x-coordinate will be -a. The y-coordinate of point B will be the same as that of point P, that is, y.

The distance between the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) = √(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2} ^{ }

Substituting the coordinates of points P and F in the distance formula,

PF = √(*x* - *a*)^{2} + (*y* - 0)^{2}

⇒ ^{ }PF = √(*x* - *a*)^{2} + *y*^{2} ……(1)

Similarly, substituting the coordinates of points P and B in the distance formula,

PB = √{*x* - (-*a*)}^{2} + (*y* - *y*)^{2}

⇒ ^{ }PB = √(*x* + *a*)^{2} ……(2)

PF = PB

⇒ √(*x* - *a*)^{2} + *y*^{2} = √(*x* + *a*)^{2}

Squaring on both sides, we get

(*x* - *a*)^{2} + *y*^{2} = (*x* + *a*)^{2}

⇒ *y*^{2} = (*x* + *a*)^{2} - (*x* - *a*)^{2}

⇒ *y*^{2 }*=* x^{2} + 2ax + a^{2}* - (*x^{2} - 2ax + a^{2}*)*

⇒ *y*^{2 }*=* x^{2} + 2ax + a^{2}* - *x^{2} + 2ax - a^{2}* *

⇒*y*^{2} = 4*ax*

The equation y^{2} = 4ax represents the given parabola.

Substituting the value of Y square as 4AX in equation 1,

*y*^{2} = 4*ax*

PF = √(*x* - *a*)^{2} + *y*^{2}

= √(*x* - *a*)^{2} + 4*ax*

= √*x*^{2} - 2*ax* + *a*^{2} + 4*ax*

= √*x*^{2} + 2*ax* + *a*^{2}

= √(*x* + *a*)^{2} = PB

Thus, the equation y^{2} = 4ax represents the parabola with the axis of symmetry along the X-axis, the vertex at the origin, the focus at (a, 0) , and directrix is x = -a.

**Equation of the parabola for case II**

By similar derivation, the equation of the parabola with the axis of symmetry along the X-axis, the vertex at the origin, the focus at (-a, 0), and directrix x= a, is y^{2} = -4ax.

**Equation of the parabola for case III**

The equation of the parabola with the axis of symmetry along the Y-axis, the vertex at the origin, the focus at (a, 0), and directrix y = -a, is *x*^{2} = 4*ay*.

**Equation of the parabola for case IV**

The equation of the parabola with the axis of symmetry along the Y-axis, the vertex at origin, the focus at (0, -a), and directrix y =a is *x*^{2} = -4*ay.*

These four equations are called the standard equations of a parabola.

If the equation of the parabola has the term *y*^{2}, then the axis of symmetry lies along the *X*-axis.

If the equation of the parabola has the term *x*^{2}, then the axis of symmetry lies along the *Y*-axis.

If the axis of symmetry lies along the *X*-axis, then the parabola opens to right if F lies on the positive side of the *X*-axis.

If the axis of symmetry lies along the *X*-axis, then the parabola opens to left if F lies on the negative side of the *X*-axis.

If the axis of symmetry lies along the *Y*-axis, then the parabola opens upwards if F lies on the positive side of the *Y*-axis.

If the axis of symmetry lies along the *Y*-axis, then the parabola opens downwards if F lies on the negative side of the *Y*-axis.

**Length of the Latus Rectum**

Consider a parabola with latus rectum AB.

Let OF = *a*

OM = OF

⇒ FM = 2*a*

FM = AC (distance of point A from directrix)

⇒ AC = 2*a*

By definition, AC = AF

⇒ AF = 2*a*

Since a parabola is symmetrical about its axis,

AF = FB

⇒ FB = 2a

⇒ Length of the Latus Rectum (AB) = AF + FB = 4a