Parabola

When you throw an object giving it some speed, it falls following a curved path, falling under the effect of uniform gravity, the path is a parabola.Many physical motions of bodies follow a curvilinear path which is in the shape of a parabola.

In mathematics, parabolas are from a family of curves called the conic section which represent curve for 2nd-degree equations. Here we shall aim at understanding the derivation of the standard formula of a parabola, the different equations of a parabola, and the properties of a parabola.

1. What is Parabola? 2. Standard Equations of a Parabola 3. Parabola Formula 4. Graphing Parabola 5. Derivation of Parabola Equation 6. Properties of Parabola 7. FAQs on Parabola

A parabola refers to an equation of a curve, such that each point on the curve is equidistant from a fixed point, and a fixed line. The fixed point is called the "focus" of the parabola, and the fixed line is called the "directrix" of the parabola. Also, an important point to note is that the fixed point does not lie on the fixed line. Thus, a parabola is mathematically defined as follows:

"A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola."

Parabola is an important curve of the conic sections of the coordinate geometry.

Parabola Equation

The general equation of a parabola is: y = a(x-h)2 + k or x = a(y-k)2 +h, where (h,k) denotes the vertex. The standard equation of a regular parabola is y2 = 4ax.

Some of the important terms below are helpful to understand the features and parts of a parabola y2 = 4ax.

• Focus: The point (a, 0) is the focus of the parabola
• Directrix: The line drawn parallel to the y-axis and passing through the point (-a, 0) is the directrix of the parabola. The directrix is perpendicular to the axis of the parabola.
• Focal Chord: The focal chord of a parabola is the chord passing through the focus of the parabola. The focal chord cuts the parabola at two distinct points.
• Focal Distance: The distance of a point ((x_1, y_1)) on the parabola, from the focus, is the focal distance. The focal distance is also equal to the perpendicular distance of this point from the directrix.
• Latus Rectum: It is the focal chord that is perpendicular to the axis of the parabola and is passing through the focus of the parabola. The length of the latus rectum is taken as LL' = 4a. The endpoints of the latus rectum are (a, 2a), (a, -2a).
• Eccentricity: (e = 1). It is the ratio of the distance of a point from the focus, to the distance of the point from the directrix. The eccentricity of a parabola is equal to 1.

Here are the formulas to find the equation of the axis, directrix, vertex, focus, and length of the latus rectum of different types of parabolas.

There are four standard equations of a parabola.

• y2 = 4ax
• y2 = -4ax
• x2 = 4ay
• x2 = -4ay

The below image presents the four standard equations and forms of the parabola.

The four standard forms are based on the axis and the orientation of the parabola. The transverse axis and the conjugate axis of each of these parabolas are different. The following are the observations made from the standard form of equations:

• Parabola is symmetric with respect to its axis. If the equation has the term with y2, then the axis of symmetry is along the x-axis and if the equation has the term with x2, then the axis of symmetry is along the y-axis.
• When the axis of symmetry is along the x-axis, the parabola opens to the right if the coefficient of the x is positive and opens to the left if the coefficient of x is negative.
• When the axis of symmetry is along the y-axis, the parabola opens upwards if the coefficient of y is positive and opens downwards if the coefficient of y is negative.

Parabola formula helps in representing the general form of the parabolic path in the plane. The following are the formulas that are used to get the parameters of a parabola.

Parabola Formulas Equation y = a(x - h)2 + k x = a(y - k)2 +h Axis of Symmetry x = h y = k Vertex (h, k) (h, k) Focus (h, k + (1/4a)) (h + (1/4a), k) Directrix y = k - 1/4a x = h - 1/4a Direction of Opening Up (a > 0) or Down (a < 0) Right (a > 0) or Left (a < 0) Length of Latus Rectum 1/a 1/a Does it Have Max or Min? Max if a < 0 Min if a > 0 Not Applicable

Consider an equation y = 3x2 - 6x + 5. For this parabola, a = 3 , b = -6 and c = 5. Here is the graph of the given quadratic equation, which is a parabola.

Direction: Here a is positive, and so the parabola opens up.

Vertex: (h, k)

h = -b/2a

= 6/(2 ×3) = 1

k = f(h)

= f(1) = 3(1)2 - 6 (1) + 5 = 2

Thus vertex is (1, 2)

Length of latus rectum = 1/a = 1/3

Focus: (h, k + 1/4a) = (1,25/12)

Axis of symmetry is x =1

Directrix: y = k-1/4a

y = 2 - 1/12 ⇒ y - 23/12 = 0

Let us consider a point P with coordinates (x, y) on the parabola. As per the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Here we consider a point B on the directrix, and the perpendicular distance PB is taken for calculations.

As per this definition of the eccentricity of the parabola, we have PF = PB (Since e = PF/PB = 1)

The coordinates of the focus is F(a,0) and we can use the coordinate distance formula to find its distance from P(x, y)

PF = (sqrt{(x - a)^2 + (y - 0)^2}) = (sqrt{(x - a)^2 + y^2})

The equation of the directtrix is x + a = 0 and we use the perpendicular distance formula to find PB.

PB = (frac{x + a}{sqrt{1^2 + 0^2}})

=(sqrt{(x + a)^2})

We need to derive the equation of parabola using PF = PB

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

Squaring the equation on both sides,

(x - a)2 + y2 = (x + a)2

x2 + a2 - 2ax + y2 = x2 + a2 + 2ax

y2 - 2ax = 2ax

y2 = 4ax

Now we have successfully derived the standard equation of a parabola.

Similarly, we can derive the equations of the other types of parabolas as:

• (b): y2 = - 4ax,
• (c): x2 = 4ay,
• (d): x2 = - 4ay.

The above four equations are the Standard Equations of Parabolas.

Here we shall aim at understanding some of the important properties and terms related to a parabola.

Tangent: The tangent is a line touching the parabola. The equation of a tangent to the parabola y2 = 4ax at the point of contact ((x_1, y_1)) is (yy_1 = 2a(x + x_1)).

Normal: The line drawn perpendicular to tangent and passing through the point of contact and the focus of the parabola is called the normal. For a parabola y2 = 4ax, the equation of the normal passing through the point ((x_1, y_1)) and having a slope of m = -y1/2a, the equation of the normal is ((y - y_1) = dfrac{-y_1}{2a}(x - x_1))

Chord of Contact: The chord drawn to joining the point of contact of the tangents drawn from an external point to the parabola is called the chord of contact. For a point ((x_1, y_1)) outside the parabola, the equation of the chord of contact is (yy_1 = 2x(x + x_1)).

Pole and Polar: For a point lying outside the parabola, the locus of the points of intersection of the tangents, draw at the ends of the chords, drawn from this point is called the polar. And this referred point is called the pole. For a pole having the coordinates ((x_1, y_1)), for a parabola y2 =4ax, the equation of the polar is (yy_1 = 2x(x + x_1)).

Parametric Coordinates: The parametric coordinates of the equation of a parabola y2 = 4ax are (at2, 2at). The parametric coordinates represent all the points on the parabola.

☛ Related Topics:

• Parabola Calculator
• Ellipse
• Hyperbola

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