Answer
405k+ views
Hint: We start solving the problem by recalling the definition of the latus rectum of the parabola as the line segment joining two ends of parabola which passes through the focus and perpendicular to the axis of the parabola. We then draw the standard parabola and then find the equation of the parabola.
Using this equation, we find the ends of the latus rectum as they were the intersection points of latus rectum and parabola. We then find the length of the latus rectum which is equal to the distance between the two end points of the latus rectum.
Complete step by step answer:
According to the problem, we need to define the latus rectum of the parabola.
We know that the latus rectum is the line segment joining two ends of the parabola which passes through the focus and perpendicular to the axis of the parabola.
Let us draw the figure representing this definition and then find the equation and length of the latus rectum.
Let us assume that ${{y}^{2}}=4ax$ is the equation of the parabola. We know that the focus of the parabola ${{y}^{2}}=4ax$ is $S\left( a,0 \right)$, vertex of the parabola is $O\left( 0,0 \right)$ and axis of the parabola is $y=0$.
Now, let us assume the ends of the latus rectum are A and B.
Let us find the equation of the latus rectum of the parabola. We know that the equation of the line perpendicular to the $y=0$ is $x=b$ but we can see that the latus rectum passes through the point $S\left( a,0 \right)$. So, we get the equation of latus rectum as $x=a$.
Now, let us substitute $x=a$ in the equation of the parabola.
So, we get ${{y}^{2}}=4a\left( a \right)=4{{a}^{2}}$.
$\Rightarrow y=\pm 2a$.
So, the ends of the latus rectum will be $A\left( a,2a \right)$ and $B\left( a,-2a \right)$.
Now, let us find the length of the latus rectum which will be the distance between the points $A\left( a,2a \right)$ and $B\left( a,-2a \right)$.
So, the length of the latus rectum = $\sqrt{{{\left( a-a \right)}^{2}}+{{\left( -2a-2a \right)}^{2}}}$.
$\Rightarrow $ Length of the latus rectum = $\sqrt{{{\left( 0 \right)}^{2}}+{{\left( -4a \right)}^{2}}}$.
$\Rightarrow $ Length of the latus rectum = $\sqrt{16{{a}^{2}}}$.
$\Rightarrow $ Length of the latus rectum = $4a$.
∴ The length of the latus rectum is $4a$.
Note: We need to know that latus rectum is a type of focal chord which is perpendicular to the axis of the parabola. We should know that the latus rectum is the only focal chord that has equal lengths above and below the axis of parabola. We should not confuse the vertex of the parabola with the focus of the parabola. Similarly, we can expect problems to find the angle subtended by the latus rectum at the vertex of parabola.
Using this equation, we find the ends of the latus rectum as they were the intersection points of latus rectum and parabola. We then find the length of the latus rectum which is equal to the distance between the two end points of the latus rectum.
Complete step by step answer:
According to the problem, we need to define the latus rectum of the parabola.
We know that the latus rectum is the line segment joining two ends of the parabola which passes through the focus and perpendicular to the axis of the parabola.
Let us draw the figure representing this definition and then find the equation and length of the latus rectum.
![seo images](https://www.vedantu.com/question-sets/3c0ac5f7-d608-450b-a64d-bb3930b68b258872090960994982422.png)
Let us assume that ${{y}^{2}}=4ax$ is the equation of the parabola. We know that the focus of the parabola ${{y}^{2}}=4ax$ is $S\left( a,0 \right)$, vertex of the parabola is $O\left( 0,0 \right)$ and axis of the parabola is $y=0$.
Now, let us assume the ends of the latus rectum are A and B.
Let us find the equation of the latus rectum of the parabola. We know that the equation of the line perpendicular to the $y=0$ is $x=b$ but we can see that the latus rectum passes through the point $S\left( a,0 \right)$. So, we get the equation of latus rectum as $x=a$.
Now, let us substitute $x=a$ in the equation of the parabola.
So, we get ${{y}^{2}}=4a\left( a \right)=4{{a}^{2}}$.
$\Rightarrow y=\pm 2a$.
So, the ends of the latus rectum will be $A\left( a,2a \right)$ and $B\left( a,-2a \right)$.
Now, let us find the length of the latus rectum which will be the distance between the points $A\left( a,2a \right)$ and $B\left( a,-2a \right)$.
So, the length of the latus rectum = $\sqrt{{{\left( a-a \right)}^{2}}+{{\left( -2a-2a \right)}^{2}}}$.
$\Rightarrow $ Length of the latus rectum = $\sqrt{{{\left( 0 \right)}^{2}}+{{\left( -4a \right)}^{2}}}$.
$\Rightarrow $ Length of the latus rectum = $\sqrt{16{{a}^{2}}}$.
$\Rightarrow $ Length of the latus rectum = $4a$.
∴ The length of the latus rectum is $4a$.
Note: We need to know that latus rectum is a type of focal chord which is perpendicular to the axis of the parabola. We should know that the latus rectum is the only focal chord that has equal lengths above and below the axis of parabola. We should not confuse the vertex of the parabola with the focus of the parabola. Similarly, we can expect problems to find the angle subtended by the latus rectum at the vertex of parabola.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 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)
Give 10 examples for herbs , shrubs , climbers , creepers
![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)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english 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)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)