Parabola: Graph, General Equation, Formulae, Latus Rectum, Focal, Normal , Tangent, Sample Questions

 A parabola is derived from conic sections. A section that is cut off, in various ways, from a right circular cone by a planne forms a conic section. The position of the cutting plane determines the shape of the coic section. The section of the right circular cone which is cut by a plane parallel to the generator of the cone is known as a parabola.

A conic section analytically can be defined as the locus of a point, which moves in such a way that its distance from some fixed point is in a constant ratio to the distance from a fixed straight line, that is not passing through the fixed point. Let us recall some important points about conic sections:

• The focus refers to the fixed point.
• The straight line that is fixed is known as the directrix.
• The constant ratio is known as eccentricity and is denoted by the letter 'e'.
• When the eccentricity is unity i.e. e = 1, the conic formed is called a parabola; when e < 1, the conic formed is called an ellipse: and when e > 1, the conic is called a hyperbola. When e = 0, the conic formed is called a circle and for e = ∞ , a pair of straight lines are formed.

Standard Equation of a Parabola

Parabola is defined as the locus of points equidistant from a fixed point (called focus S) and a fixed-line (called directrix MZ).

• For horizontal parabola

• Let us consider

1. Origin (0,0) as the parabola's vertex A, 
2. Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,  
3. P(x,y) as the moving point. 

• Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola. 
• The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ. 
• The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure. 
• By definition PM = PS 

=> MP² = PS² 

• So, (a + x)² = (x - a)² + y².
• Hence, we can get the equation of horizontal parabola as y² = 4ax.

• For vertical parabola

• Let us consider

1. Origin (0,0) as the parabola's vertex A
2. Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
3. P(x,y) as any moving point 

• Let us now draw a perpendicular SZ from S to the directrix. 
• Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ. 
• The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
• By definition PM = PS 

=> MP² = PS²

So, (b + y)² = (y - b)² + x²

• As a result, the vertical parabola equation is x²= 4by.

Discover about the Chapter video:

Conic Sections Detailed Video Explanation:

General Equation of any Parabola

Let us now find out the equation of a parabola when the directrix is any line and the focus is any point. 

Let us assume (h,k) is the focus S and the line Ix + my + n = 0 is the equation of the directrix ZM of the parabola. Let us take (x,y) as the coordinate of any point P on the parabola. The relationship PS = distance of P from ZM thus yields - 

(x- h)² + (y - k)² = (Ix + my + n)² / (l²+ m²)

=› (Ix - my)² + 2gx + 2fy + d = 0

This is the general equation of any parabola. The second-degree terms in the equation of the parabola constitute a perfect square. The converse will be also true, i.e. If the second-degree components in an equation form a perfect square, the equation represents a parabola, unless the equation represents two parallel straight lines.

Note: The general second degree polynomial function i.e. ax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a parabola if Δ ≠ 0 and h² = ab (where Δ = abc+ 2fgh - af² - bg² - ch²).

Latus Rectum of a Parabola

Latus rectum is known as the chord that passes through the focus and is perpendicular to the axis of the parabola.

We will now put x = a as the latus rectum that passes through focus (a,0) to find the endpoints of latus rectum LL' of the parabola y² = 4ax.

Therefore we have,

y² = 4a²

=> y= ±2a

As a result, the endpoints are (a, 2a) and (a, -2a).

Also LSL' = |2a - (-2a)| = 4a = length of double ordinate through the focus S.

Note: When the latus rectum of two parabolas is equal, then these parabolas are said to be equal.

Important points to remember about the standard parabolas

Note: 

1. The following transformations can be used to interchange the points and lines of two parabolas -

          x ↔ y, a ↔ b, (α,β) ↔ (β,α )

2. if a > 0 & a < 0 then the parabola will be forward opening and backward opening respectively.
3. if b > 0 & b < 0 then the parabola will be upward opening and downward opening respectively.

Important points to remember about any shifted Parabola

The forms of the horizontal and vertical parabolas having the vertex at (h,k) can be obtained by shifting the origin at (h,k).

Parametric Form of a Parabola

Let us assume that the tangent equation to the parabola y² = 4ax-- (i)

is y = mx + c …..(ii)

The equation (mx + c)² = 4ax gives us the x-cordinate of the points of intersection of (i) and (ii). 

However, the criterion for the straight line (ii) is that it must meet the parabola at coinciding locations. Hence, discriminant should be zero

=> (mc - 2a)² = m²c²

=> c = a/m

As a result, y = mx + a/m gives us the equation of the tangent to the parabola y² = 4ax, where ‘m’ can take any value.

So, the equation (mx + c)² = 4ax has now become (mx - a/m)² = 0

=> x = a/m²

and y² = 4ax

=> y = 2a/m

Thus the tangent y = mx + a/m contacts the parabola at these point - (a/m², 2a/m)

Taking 1/m= t where t is a parameter, i.e., it can vary from point to point on any curve. The parabola y² = 4ax can be represented in parametric form by the co-ordinate (at²,2at) and we refer to it as point ‘t' on the parabola.

Focal Chord of a Parabola

Any chord which passes through the focus is known as a focal chord of the parabola.

Let y2 = 4ax be the equation of a parabola and point P be at (at²,2at) on the parabola. Let us assume that the coordinates of the extremity Q of the focal chord passing through point P are (at₁²,2at₁).

Then, PS and SQ (where S(a,0) is the focus), have same slopes

=> 2at - 0/at² - a = 2at₁ - 0/at₁² - a

=> tt₁² - t = t₁t² - t₁  => (tt₁ + 1) (t₁ - t) = 0

So, t₁= -1/t, that means the point Q can be written as (a/t²,-2a/t).

So, for simplicity, the extremities of any focal chord of the parabola y2 = 4ax can be assumed as the points t and -1/t.

Focal Distance of a Point 

A point P on the parabola, y² = 4ax has a focal distance of PS, which is the distance between the point P and the focus S. So, the focal distance of P becomes - y² = 4ax

=> PM = ZN = PS = AN + AZ = a + x

or

=> PS = a+ at² = a(1 + t²)

Position of any point with respect to a Parabola

Consider the parabola y² = 4ax. Let us assume (x₁,y₁) is any given point P on the parabola and y₁² - 4ax₁ = 0, then that point will lie on the parabola. However, when y₁²-4ax₁ ≠ 0, y-coordinate PM is plotted to meet the curve at point L. 

Now, point P would lie outside of the parabola if -

LM < PM, that is, PM² - LM² > 0.

Now, PM² = y₁² and LM² = 4ax₁ by reference to the coordinates of point ‘L’ which satisfies the equation of the given parabola. As a result, the condition for point P to lie outside of the parabola becomes y₁² - 4ax₁ > 0.

In the same way, we can conclude that the condition for point P to lie inside of the parabola, is y₁² - 4ax₁ < 0.

Tangent to a Parabola

A line that touches any curve at exactly one point is called a tangent to that curve.

The tangent at the point (x₁, y₁)

Let us assume the equation of the standard parabola is y² = 4ax. As a result, the value of dy/dx at point P(x₁,y₁) is 2a/y1  and thus the equation of the tangent at P is 

y - y₁= 2a/y₁(x-x₁)

i.e. yy₁ = 2a(x-x₁) + y₁²

=> yy₁= 2a(x+x₁)

In other words, we can write the equation of chord joining the points P (x₁,y₁) and Q (x₂,y₂) on the parabola y₂ = 4ax. Equation of the chord is

=> (x-x₁)/(x₂ - x₁)= (y- y₁)/(y₂ - y₁)

(x-x₁)/(x₂-x₁)= (y-y₁)(y₂+y₁)/ y₂²-y₁² = (y-y₁)(y₂ + y₁)/ 4a(x₂ - x₁) or 4a(x -x₁) = (y -y₁) (y₂ + y₁) .

When the two points P and Q tend to coincide, y₂ → y₁ and then the line PQ will be the tangent to the given parabola. Its equation is 4a(x - x₁) = (y - y₁)(2y₁) or yy₁ = 2a(x + x₁)

Tangent in terms of m

Let us assume that the tangent equation to the parabola y² = 4ax...(i)

is y = mx + c.....(i)

The abscissa of the intersection points of (i) and (ii) are given by the equation:

(mx+c)²= 4ax  

However, the criterion for the straight line (ii) to contact the parabola is that it must do so at coinciding locations.

=> (mc - 2a)² = m²c².... (iii)

=> c = a/m

Hence, y = mx + a/m is a tangent to the parabola y² = 4ax, for all the possible values of ‘m’.

Now, the equation (mx +c)² = 4ax becomes (mx - a/m)² = 0 . 

x = a/m² and 

y² = 4ax

=> y = 2a/m

As a result, the tangent y = mx + a/m has (a/m²,2a/m) as contact point.

Tangent at the point ‘t’

The tangent equation at ‘t' is ty = x + at² .

(at₁t₂, a(t₁+t₂)) is the point of intersection of the tangents at 't₁' and 't₂' .

Tangent equations from an External Point

Let y² = 4ax be the equation of a parabola and (x₁,y₁) be any external point P . Now, the equation of the tangents can be written as :

SS₁ = T², where S = y² - 4ax, S₁ = y₁² - 4ax₁, T = yy₁ - 2a(x + x₁).

PQ is the chord of contact of the tangents if the tangents from the exterior point (x₁,y₁) touch the parabola at P and Q.

Chord of Contact

The equation for the chord of contact of the tangent that is drawn from a point (x₁,y₁) to a parabola y² = 4ax is T = 0, i.e. yy₁ - 2a(x+x₁) = 0.

Note: The equation for the chord of the parabola y²= 4ax with its midpoint at (x₁,y₁) is T=S₁

i.e. yy₁ - 2a(x + x₁) = y₁² - 4ax₁ .

or 

yy₁-2ax = y₁²-2ax

Normal to a Parabola

A line that is perpendicular to the tangent of any curve at its point of contact to the curve, is called a normal to that particular curve.

Normal at the point (x₁,y₁)

The equation of the tangent at the point (x₁,y₁) is yy₁ = 2a( x + x₁). 

Since,

Slope of the tangent = 2a/y₁

The slope of the normal is -y₁/2a. 

Also, it will pass through the point (x₁,y₁)

As a result, the equation of the normal at a point is (y - y₁) = -y₁/2a(x - x₁)......(i)

Normal in terms of m

In equation (i), put -y₁/2a = m so that y₁ = -2am and x₁ = y₁²/4a = am², then the equation becomes 

y = mx - 2am - am³…….(ii)

where, ‘m’ is the slope. The equation (ii) is the equation of the normal at the point (am²,-2am) of the parabola. am³

Note:  

• If this normal passes through any point (h, k), then k = mh-2am-am³ for any given parabola and any given point (h,k), then this cubic variable in ‘m' will have three roots, namely m₁, m₂, and m₃, which means that three normals can be drawn from (h,k) to the parabola whose slopes are m₁, m₂, and m₃ respectively. 

For this cubic function, we have 

m₁ + m₂ + m₃ = 0, m₁m₂ + m₂m₃ + m₃m₁ = (2a-h) / a,

m₁m₂m₃ = -k/a. 

If we add an extra condition to the normals drawn from any point (h, k) to a specified parabola, such as y² =4ax, we may get the locus of the parabola by elimination of m₁, m₂, and m₃ from the four relations between them.

• Because the sum of the roots, m₁, m₂, and m₃, is zero; the sum of the ordinates of the feet of the normals from any given position is also zero. These points are known as Co-Normal Points.

Normal at the point ‘t’

y = -tx + 2at + at³ is the normal which is perpendicular to the tangent at the point (at²,2at)

Note:

• If the normal at t₁ point meets the parabola again at point t₂, then t₂= -t₁ - 2/t₁
• Point of intersection of the normals to the parabola y² = 4ax at (at₁²,2at₁) and (at₂²,2at₂) is (2a + a (t₁² + t₂² + t₁t₂), - at₁t₂(t₁+ t₂)).

Sample Questions of Parabola