Answer
397.2k+ views
Hint: In this question, we have to find the parametric equation of a parabola. As we know, a parabola is a symmetrical plane or a curve that is formed by the intersection of the cone with a plane parallel to its side. Also, a parametric equation is a type of equation that employs an independent variable called a parameter and in which dependent variables are continuous functions of the parameter. Therefore, in this problem, we have to find the parametric equation of the parabola.
Complete step-by-step solution:
According to the question, we have to find the parametric equation of a parabola. A parabola is a symmetrical plane that is formed by the intersection of the cone with a plane parallel to its side.
So, we will solve four different cases of the parabola.
Case 1: Now, we know the standard equation of the parabola is ${{y}^{2}}=-4ax$ ----- (1)
Therefore, the graph of the equation ${{y}^{2}}=-4ax$ is as follows:
Now, we will let $x=-a{{t}^{2}}$ and $y=2at$ , and put the value of x and y in the equation (1), we get
${{\left( 2at \right)}^{2}}=-4a\left( -a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{y}^{2}}=-4ax$ , its parametric equations are $x=-a{{t}^{2}}$ and $y=2at$.
Case 2: Now, we know the standard equation of the parabola is ${{y}^{2}}=4ax$ ----- (2)
Therefore, the graph of the equation ${{y}^{2}}=4ax$ is as follows:
Now, we will let $x=a{{t}^{2}}$ and $y=2at$ , and put the value of x and y in the equation (2), we get
${{\left( 2at \right)}^{2}}=4a\left( a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{y}^{2}}=4ax$ , its parametric equations are $x=a{{t}^{2}}$ and $y=2at$.
Case 3: Now, we know the standard equation of the parabola is ${{x}^{2}}=4ay$ ----- (3)
Therefore, the graph of the equation ${{x}^{2}}=4ay$ is as follows:
Now, we will let $x=2at$ and $y=a{{t}^{2}}$ , and put the value of x and y in the equation (3), we get
${{\left( 2at \right)}^{2}}=4a\left( a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{x}^{2}}=4ay$ , its parametric equations are $x=2at$ and $y=a{{t}^{2}}$.
Case 4: Now, we know the standard equation of the parabola is ${{x}^{2}}=-4ay$ ----- (4)
Therefore, the graph of the equation ${{x}^{2}}=-4ay$ is as follows:
Now, we will let $x=2at$ and $y=-a{{t}^{2}}$ , and put the value of x and y in the equation (4), we get
${{\left( 2at \right)}^{2}}=-4a\left( -a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{x}^{2}}=-4ay$ , its parametric equations are $x=2at$ and $y=-a{{t}^{2}}$.
Therefore, we have found the 4 parametric equations of the parabola.
Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical error. Do to forget the standard form of the equation of the parabola. Also, do all the four cases, to get the required result for the solution.
Complete step-by-step solution:
According to the question, we have to find the parametric equation of a parabola. A parabola is a symmetrical plane that is formed by the intersection of the cone with a plane parallel to its side.
So, we will solve four different cases of the parabola.
Case 1: Now, we know the standard equation of the parabola is ${{y}^{2}}=-4ax$ ----- (1)
Therefore, the graph of the equation ${{y}^{2}}=-4ax$ is as follows:
![seo images](https://www.vedantu.com/question-sets/c0be7eb4-d48c-4837-9ea6-7db3ebfea0981631646418783612201.png)
Now, we will let $x=-a{{t}^{2}}$ and $y=2at$ , and put the value of x and y in the equation (1), we get
${{\left( 2at \right)}^{2}}=-4a\left( -a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{y}^{2}}=-4ax$ , its parametric equations are $x=-a{{t}^{2}}$ and $y=2at$.
Case 2: Now, we know the standard equation of the parabola is ${{y}^{2}}=4ax$ ----- (2)
Therefore, the graph of the equation ${{y}^{2}}=4ax$ is as follows:
![seo images](https://www.vedantu.com/question-sets/d7067a55-2bba-479b-9147-f2d88ba33f90743654158913147460.png)
Now, we will let $x=a{{t}^{2}}$ and $y=2at$ , and put the value of x and y in the equation (2), we get
${{\left( 2at \right)}^{2}}=4a\left( a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{y}^{2}}=4ax$ , its parametric equations are $x=a{{t}^{2}}$ and $y=2at$.
Case 3: Now, we know the standard equation of the parabola is ${{x}^{2}}=4ay$ ----- (3)
Therefore, the graph of the equation ${{x}^{2}}=4ay$ is as follows:
![seo images](https://www.vedantu.com/question-sets/37c7d4c8-da61-4045-b234-95b402ac7f397963700430844854158.png)
Now, we will let $x=2at$ and $y=a{{t}^{2}}$ , and put the value of x and y in the equation (3), we get
${{\left( 2at \right)}^{2}}=4a\left( a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{x}^{2}}=4ay$ , its parametric equations are $x=2at$ and $y=a{{t}^{2}}$.
Case 4: Now, we know the standard equation of the parabola is ${{x}^{2}}=-4ay$ ----- (4)
Therefore, the graph of the equation ${{x}^{2}}=-4ay$ is as follows:
![seo images](https://www.vedantu.com/question-sets/513f1170-e2f9-4887-ab91-f5cbfe2ebaad2735278669076621504.png)
Now, we will let $x=2at$ and $y=-a{{t}^{2}}$ , and put the value of x and y in the equation (4), we get
${{\left( 2at \right)}^{2}}=-4a\left( -a{{t}^{2}} \right)$
On further solving, we get
\[4{{a}^{2}}{{t}^{2}}=4{{a}^{2}}{{t}^{2}}\]
Thus, we see that both the LHS and RHS are equal to each other, which implies for the parabolic equation ${{x}^{2}}=-4ay$ , its parametric equations are $x=2at$ and $y=-a{{t}^{2}}$.
Therefore, we have found the 4 parametric equations of the parabola.
Note: While solving this problem, do mention all the steps properly to avoid confusion and mathematical error. Do to forget the standard form of the equation of the parabola. Also, do all the four cases, to get the required result for the solution.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)