# Parabola whose Vertex at a given Point and Axis is Parallel to x-axis

We will discuss how to find the equation of the parabola whose vertex at a given point and axis is parallel to x-axis.

Let A (h, k) be the vertex of the parabola, AM is the axis of the parabola which is parallel to x-axis. The distance between the vertex and focus is AS = a and let P (x, y) be any point on the required parabola.

Now we shift the origin of co-ordinate system at A. Draw two mutually perpendicular straight lines AM and AN through the point A as x and y-axes respectively.

According to the new co-ordinate axes (x', y ') be the co-ordinates of P. Therefore, the equation of the parabola is (y')$$^{2}$$ = 4ax' (a > 0) …………….. (i)

Therefore, we get,

AM = x' and PM = y'

Also, OR = h, AR = k, OQ = x, PQ = y

Again, y = PQ

= PM + MQ

= PM + AR

= y' + k

Therefore, y' = y - k

And, x = OQ = OR + RQ

= OR + AM

= h + x'

Therefore, x' = x - h

Now putting the value of x' and y' in (i) we get

(y - k)$$^{2}$$ = 4a(x - h), which is the equation of the required parabola.

The equation (y - k)$$^{2}$$ = 4a(x - h) represents the equation of a parabola whose co-ordinate of the vertex is at (h, k), the co-ordinates of the focus are (a + h, k), the distance between its vertex and focus is a, the equation of directrix is x - h = - a or, x + a = h, the equation of the axis is y = k, the axis is parallel to positive x-axis, the length of its latus rectum = 4a, co-ordinates of the extremity of the latus rectum are (h  + a, k + 2a) and (h + a, k - 2a) and the equation of tangent at the vertex is x = h.

Solved example to find the equation of the parabola with its vertex at a given point and axis is parallel to x-axis:

Find the axis, co-ordinates of vertex and focus, length of latus rectum and the equation of directrix of the parabola y$$^{2}$$ + 4x + 2y - 11 = 0.

Solution:

The given parabola y$$^{2}$$ + 4x + 2y - 11 = 0.

y$$^{2}$$ + 4x + 2y - 11 = 0

⇒ y$$^{2}$$ + 2y + 1 - 1 + 4x - 11 = 0

⇒ (y + 1)$$^{2}$$ = -4x + 12

⇒ {y - (-1)}$$^{2}$$ = -4(x - 3)

⇒ {y - (-1)}$$^{2}$$ = 4 ∙ (-1) (x - 3) …………..(i)

Compare the above equation (i) with standard form of parabola (y - k)$$^{2}$$ = 4a(x - h), we get, h = 3, k = -1 and a = -1.

Therefore, the axis of the given parabola is along parallel to negative x-axis and its equation is y = - 1 i.e., y + 1 = 0.

The co-ordinates of its vertex are (h, k) i.e., (3, -1).

The co-ordinates of its focus are (h + a, k) i.e., (3 - 1, -1) i.e., (2, -1).

The length of its latus rectum = 4 units

The equation of its directrix is x + a = h i.e., x - 1 = 3 i.e., x - 1 - 3 = 0 i.e., x - 4 = 0.

● The Parabola

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### 2nd Grade Place Value | Definition | Explanation | Examples |Worksheet

Sep 14, 24 04:31 PM

The value of a digit in a given number depends on its place or position in the number. This value is called its place value.

2. ### Three Digit Numbers | What is Spike Abacus? | Abacus for Kids|3 Digits

Sep 14, 24 03:39 PM

Three digit numbers are from 100 to 999. We know that there are nine one-digit numbers, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. There are 90 two digit numbers i.e., from 10 to 99. One digit numbers are ma

3. ### Worksheet on Three-digit Numbers | Write the Missing Numbers | Pattern

Sep 14, 24 02:12 PM

Practice the questions given in worksheet on three-digit numbers. The questions are based on writing the missing number in the correct order, patterns, 3-digit number in words, number names in figures…

4. ### Comparison of Three-digit Numbers | Arrange 3-digit Numbers |Questions

Sep 13, 24 02:48 AM

What are the rules for the comparison of three-digit numbers? (i) The numbers having less than three digits are always smaller than the numbers having three digits as: