Problems on Parabola

We will learn how to solve different types of problems on parabola.

1. Find the vertex, focus, directrix, axis and latusrectum of the parabola y\(^{2}\) - 4x - 4y = 0

Solution:

The given equation of the parabola is y\(^{2}\) - 4x - 4y = 0

⇒ y\(^{2}\) - 4y = 4x

⇒ y\(^{2}\) - 4y + 4 = 4x + 4, (Adding 4 on both sides)

⇒ (y - 2)\(^{2}\) = 4(x  + 1) ……………………………….. (i)

Shifting the origin to the point (-1, 2) without rotating the axes and denoting the new coordinates with respect to these axes by X and Y, we have

x = X + (-1), y = Y + 2 ……………………………….. (ii)

Using these relations equation (i), reduces to

Y\(^{2}\) = 4X……………………………….. (iii)

This is of the form Y\(^{2}\) = 4aX. On comparing, we get 4a = 4 ⇒ a = 1.

The coordinates of the vertex with respect to new axes are (X = 0, Y = 0)

So, coordinates of the vertex with respect to old axes are (-1, 2), [Putting X= 0, Y = 0 in (ii)].

The coordinates of the focus with respect to new axes are (X = 1, Y = 0)

So, coordinates of the focus with respect to old axes are (0, 2), [Putting X= 1, Y = 0 in (ii)].

Equation of the directrix of the parabola with respect to new axes in X = -1

So, equation of the directrix of the parabola with respect to old asex is x = -2, [Putting X = -1, in (ii)].

Equation of the axis of the parabola with respect to new axes is Y = 0.

So, equation of axis with respect to old axes is y = 2, [Putting Y = 0, in (ii)].

The length of the latusrectum is 4 units.

 

2. Find the point on the parabola y\(^{2}\) = 12x at which the ordinate is double the abscissa. 

Solution: 

The given parabola is y\(^{2}\) = 12x.

Now, let (k, 2k) be the co-ordinates of the required point (k ≠ 0).

Since the point lies (k, 2k) on the parabola y\(^{2}\) = 12x,

Therefore, we get,

 (2k)\(^{2}\) = 12k

⇒ 4k\(^{2}\) = 12k     

⇒ k = 3 (Since, k ≠ 0, ).

Therefore, the co-ordinates of the required point are (3, 6).

 

3. Write the parametric equation of the parabola (x + 2)\(^{2}\) = - 4(y + 1).

Solution:

The given equation of the parabola is (x + 2)\(^{2}\) = - 4(y + 1).

Then parametric equation of the parabola (x + 2)\(^{2}\) = - 4(y + 1) are

x + 2 = 2t and y + 1 = -t\(^{2}\)

⇒ x = 2t – 2 and y = -t\(^{2}\) – 1.

 

4. Find the equation of the parabola whose co-ordinates of vertex and focus are (-2, 3) and (1, 3) respectively.

Solution:           

According to the problem, the ordinates of vertex and focus are equal hence, the axis of the required parabola is parallel to x-axis. Again,

a = abscissa of focus - abscissa of vertex

⇒ a = 1 - (- 2) = 1 + 2 = 3.

Therefore, the equation of the required parabola is

 (y - β)\(^{2}\) = 4a (x - α)                 

⇒ (y - 3)\(^{2}\) = 4 . 3(x + 2)

⇒ y\(^{2}\) - 6y + 9 = 12x + 24            

⇒ y\(^{2}\) - 6y - 12x = 15.

● The Parabola




11 and 12 Grade Math 

From Problems on Straight Lines to HOME PAGE




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.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Converting Fractions to Decimals | Solved Examples | Free Worksheet

    Apr 28, 25 01:43 AM

    Converting Fractions to Decimals
    In converting fractions to decimals, we know that decimals are fractions with denominators 10, 100, 1000 etc. In order to convert other fractions into decimals, we follow the following steps:

    Read More

  2. Expanded Form of a Number | Writing Numbers in Expanded Form | Values

    Apr 27, 25 10:13 AM

    Expanded Form of a Number
    We know that the number written as sum of the place-values of its digits is called the expanded form of a number. In expanded form of a number, the number is shown according to the place values of its…

    Read More

  3. Converting Decimals to Fractions | Solved Examples | Free Worksheet

    Apr 26, 25 04:56 PM

    Converting Decimals to Fractions
    In converting decimals to fractions, we know that a decimal can always be converted into a fraction by using the following steps: Step I: Obtain the decimal. Step II: Remove the decimal points from th…

    Read More

  4. Worksheet on Decimal Numbers | Decimals Number Concepts | Answers

    Apr 26, 25 03:48 PM

    Worksheet on Decimal Numbers
    Practice different types of math questions given in the worksheet on decimal numbers, these math problems will help the students to review decimals number concepts.

    Read More

  5. Multiplication Table of 4 |Read and Write the Table of 4|4 Times Table

    Apr 26, 25 01:00 PM

    Multiplication Table of Four
    Repeated addition by 4’s means the multiplication table of 4. (i) When 5 candle-stands having four candles each. By repeated addition we can show 4 + 4 + 4 + 4 + 4 = 20 Then, four 5 times

    Read More