Let us imagine a basketball game. As a player shoots the ball, the ball follows a curved path before landing in the hoop. The path that the ball takes is a parabola.
A parabola is a U-shaped curve that is also found in the path of water in a fountain or the shape of satellite dishes.
Mathematically, it is the graph of a quadratic function in two variables, y = ax2 + bx + c, that can be defined as the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. The vertex is the point where the parabola is closest to the directrix and where it changes direction. The y = ax2 + bx + c is thus a parabolic function.
The shape of a parabola is found in the trajectory of rockets and the arc of water from the fountains. It has many applications, such as in headlights, bridge cables, and solar panels.
Parts
Vertex: It is the turning point of the parabola and also the highest or lowest point, depending on whether it opens upward or downward. The vertex of the above parabola is at (0, 0).
Axis of Symmetry: It is the dashed line passing through the vertex of the parabola and dividing it into two symmetric halves.
Focus: Has 1 focus, with coordinates (0, a)
Directrix: It is the line drawn perpendicular to the axis of the parabola. Here, the line parallel to the x-axis and passing through the point (0, -a) is the directrix.
Focal Chord: It is the chord that intersects the parabola at two distinct points and passes through the focus.
Latus Rectum: It is the focal chord of the parabola that is perpendicular to its axis and passes through the focus. The endpoints of the latus rectum are (2a, a) and (-2a, a), and its length is 4a.
Focal Distance: It is the distance from a point (x1, y1) on the parabola to the focus. The focal distance is also equal to the perpendicular distance from this point to the directrix.
Eccentricity: It is the ratio of the distance between any point on the curve and the focus to the distance between that point and the directrix. For a parabola, the eccentricity is always equal to 1.
Equations
The equation of a parabola depends on its orientation, the position of its vertex, and whether it is centered at the origin or elsewhere.
Standard Form
A horizontal parabola is a parabola that opens sideways, either to the left or to the right.
y2 = 4ax
Here,
The equation of the axis of symmetry is y = 0
The equation of directrix is x = -a
Vertex at (0, 0)
Focus at (a, 0)
The length of the latus rectum is 4a
This parabola opens to the right, forming a ‘⊂’ shaped curve.
Derivation: y2 = 4ax
Let us derive the above equation.
Let us consider a point P on the parabola with the coordinates (x, y)
By the definition of a parabola, the distance of point P from the focus F and the distance of point P from the directrix is the same.
Now, we will draw a perpendicular line from point P that intersects the directrix at B, as shown.
As we know, the eccentricity of a parabola is always 1
We have e = 1
⇒ ${\dfrac{PF}{PB}=1}$
⇒ PF = PB
Since the coordinates of the focus are F(a,0), we can determine its distance from the point P (x, y)
If the parabola is of the form y2 = 4ax, then its parametric coordinates are (at2, 2at). It represents all the points on the parabola.
y2 = -4ax
Here,
The equation of the axis of symmetry is y = 0
The equation of directrix is x = a
Vertex at (0, 0)
Focus at (-a, 0)
The length of the latus rectum is 4a
This parabola opens to the left, forming a ‘⊃’ shaped curve.
When a parabolaopens either upward or downward, it is called a vertical parabola.
x2 = 4ay
Here,
The equation of the axis of symmetry is x = 0
The equation of directrix is y = -a
Vertex at (0, 0)
Focus at (0, a)
The length of the latus rectum is 4a
This parabola opens upward, forming a ‘U’ shaped curve.
x2 = -4ay
Here,
The equation of the axis of symmetry is x = 0
The equation of directrix is y = a
Vertex at (0, 0)
Focus at (0, -a)
The length of the latus rectum is 4a
This parabola opens downward, forming a ‘⋂’ shaped curve.
As we derived the equation for the parabola y2 = 4ax, we can similarly derive the equations for other types of parabolas.
Below is a summary of the formulas to determine the axis, directrix, vertex, focus, and length of the latus rectum for parabolas in different orientations when expressed in standard form:
Find the focus from the given equation of the parabola: a) y2 = 20x b) x2 = -4y
Solution:
a) Given y2 = 20x …..(i) As we know, the standard equation of the parabola is y2 = 4ax …..(ii) Comparing the equations (i) and (ii), a = 5 Thus, the focus is at (a, 0) = (5, 0) b) Given x2 = -4y …..(i) As we know, the standard equation of the parabola is x2 = -4ay …..(ii) Comparing the equations (i) and (ii), a = 1 Thus, the focus is at (0, -a) = (0, -1)
The equation of a parabola is y2 = 16x. Find the length of the latus rectum, focus, and vertex.
Solution:
As we know, The equation of a parabola is y2 = 16x Now, 4a = 16 a = 4 Thus, The length of the latus rectum = 4a = 4 × 4 = 16 Focus = (a, 0) = (4, 0) Vertex = (0, 0)
Vertex Form
Another useful form of writing the equation of a parabola is the vertex form, which highlights the vertex of the parabola when graphed.
The vertex form of the equation of a vertical parabola is given by:
y = a(x – h)2 + k
Here,
The equation of the axis of symmetry is x = h
The equation of directrix is ${y=k-\dfrac{1}{4a}}$
Vertex at (h, k)
Focus at ${\left( h,k+\dfrac{1}{4a}\right)}$
The length of the latus rectum is ${\dfrac{1}{a}}$
Derivation of Quadratic Function: y = ax2 + bx + c
By rearranging the vertex form of the parabola, we get
y = a(x – h)2 + k
⇒ y = a(x2 – 2hx + h2) + k
⇒ y = ax2 – 2ahx + ah2 + k
⇒ y = ax2 + (-2ah)x + (ah2 + k)
If we replace (-2ah) with b and (ah2 + k) with c, the standard form of a parabola is also written as:
y = ax2 + bx + c, here, a ≠ 0, b, and c are constants.
The value of ‘a’ determines whether the parabola opens upward or downward:
If a > 0, then the parabola opens upward.
If a < 0, then the parabola opens downward.
The vertex form of the equation of a horizontalparabola is given by:
x = a(y – k)2 + h
Here,
The equation of the axis of symmetry is y = k
The equation of directrix is ${x=h-\dfrac{1}{4a}}$
Vertex at (h, k)
Focus at ${\left( h+\dfrac{1}{4a},k\right)}$
The length of the latus rectum is ${\dfrac{1}{a}}$
Here is a table listing all the formulas that are used to find the axis, directrix, vertex, focus, and length of the latus rectum of the parabolas whose vertex is at (h, k):
As we obtained the quadratic form for the parabola y = ax2 + bx + c, we can similarly derive the equations for vertical parabolas.
The quadratic equation of a vertical parabola is written as x = ay2 + by + c. Here, a ≠ 0, b, and c are constants.
Graphing
To graph a parabola, we find the vertex of the parabola and the axis of symmetry, and then, sketch the curve.
For the equation of the parabola y = ax2 + bx + c, the x-coordinate for the vertex is ${h=-\dfrac{b}{2a}}$
By substituting this value in the equation, the y-coordinate for the vertex is: k = a(h)2 + b(h) + c
Let us graph the parabola y = x2
Comparing the given equation with the standard form of a parabola y = ax2 + bx + c, we get
Now, plotting the given quadratic equation y = x2 on the graph, we get
Graph the parabola x = 3(y – 2)2 + 4. Find the focus, vertex, and length of the latus rectum.
Solution:
Given x = 3(y – 2)2 + 4 …..(i) As we know, the standard form of the parabola is x = a(y – k)2 +h …..(ii) Comparing the equations (i) and (ii), we get a = 3, h = 4, and k = 2 Since a is positive, the parabola opens to the right. Now, The vertex is at (h, k) = (4, 2) The length of the latus rectum = ${\dfrac{1}{a}}$ = ${\dfrac{1}{3}}$ Focus = ${\left( h+\dfrac{1}{4a},k\right)}$ = ${\left( 4+\dfrac{1}{4\times 3},2\right)}$ = ${\left( \dfrac{49}{12},2\right)}$ The axis of symmetry is y = k ⇒ y = 2 Directrix is ${x=h-\dfrac{1}{4a}}$ ⇒ ${x=4-\dfrac{1}{4\times 3}}$ ⇒ ${x=\dfrac{47}{12}}$ Now, plotting the given quadratic equation y = x2 on the graph, we get the required graph.
Find the equation of the parabola whose graph is shown below.
Solution:
The graph has a vertex at (2, 3). Hence, the equation of the parabola in vertex form is written as y = a(x – 2)2 + 3 We now use the y-intercept at (0,−1) to find the coefficient a -1 = a(0 – 2) + 3 Now, we will solve for ‘a’ a = 2 Thus, the equation of the parabola, whose graph is shown above, is y = 2(x – 2)2 + 3
Properties
Tangent
A line that touches the parabola at a single point is called its tangent.
If the equation of the parabola is y2 = 4ax, the equation of its tangent at the point of contact (x1, y1) is yy1 = 2a(x + x1)
Normal
The line drawn perpendicular to the tangent that passes through both the point of contact and the focus of the parabola is called the normal.
If the parabola is of the form y2 = 4ax and the normal passes through the point (x1, y1) with a slope of ${m=-\dfrac{y_{1}}{2a}}$, then its equation is ${\left( y-y_{1}\right) =-\dfrac{y_{1}}{2a}\left( x-x_{1}\right)}$
Chord of Contact
The chord of contact is formed by joining the points of contact of the tangents drawn from an external point to the parabola.
If a point (x1, y1) is outside the parabola, the equation of the chord of contact is yy1 = 2x(x + x1)
Here,
P is any point with coordinates (x1, y1)
CD and EF are tangents
AB is chord of contact
BG is normal
Problem: Finding the equation of a parabola when the VERTEX and DIRECTRIX are given
If the vertex of the parabola is at (3, 2) and the directrix of the parabola is y = 5, find the equation of the parabola.
Solution:
The equation of the parabola in the vertex form is generally written as (x – h)2 = 4p(y – k) …..(i) Here, (h, k) = vertex of the parabola p = distance from the vertex to the directrix (or the focus) Given the vertex is (3, 2) and the directrix is y = 5 Here, h = 3, k = 2, and p = 2 – 5 = -3 Substituting the values in the equation (i), we get (x – 3)2 = 4(-3)(y – 2) ⇒ x2 – 6x + 9 = -12y + 24 ⇒ 12y = -x2 + 6x – 9 + 24 ⇒ 12y = -x2 + 6x + 15 ⇒ ${y=-\dfrac{1}{12}x^{2}+\dfrac{6}{12}x+\dfrac{15}{12}}$ ⇒ ${y=-\dfrac{1}{12}x^{2}+\dfrac{1}{2}x+\dfrac{5}{4}}$ Thus, the equation of the parabola is ${y=-\dfrac{1}{12}x^{2}+\dfrac{1}{2}x+\dfrac{5}{4}}$
