Answer

Verified

63.3k+ views

Hint: The form of the parabola to be used in the questions is \[{{\left( x-{{x}_{1}} \right)}^{2}}=4a\left( y-{{y}_{1}} \right)\].

Complete step-by-step answer:

The vertex and the latus rectum of a parabola are given as $\left( a,b \right)$ and \[l\] respectively in the question.

Since the axis of the parabola is along the positive direction of the $y-$axis, we can figure out that the form of the required parabola would be \[{{x}^{2}}=4ay\]. Latus rectum is indicated by LR and the given point of the vertex is termed as A. We can represent the details as shown in the figure below.

The vertex, A is $\left( a,b \right)$, so we can write the equation for the parabola as,

\[{{\left( x-a \right)}^{2}}=4c\left( y-b \right)\ldots \ldots \ldots (i)\]

Since the coordinate of the vertex is \[a\], the term $c$ has been used in the equation above.

From the figure, we can see that the latus rectum is perpendicular to the axis of the parabola and is represented in the figure as LR. We know that the length of the latus rectum for the form of parabola, \[{{x}^{2}}=4ay\] is \[4c\]. Also, it is already given to us in the question as \[l\].

Therefore, we can relate the data and we can write the term \[4c=l\].

After substituting this relation in equation \[(i)\], we get the equation of the parabola as,

\[{{\left( x-a \right)}^{2}}=l\left( y-b \right)\]

Hence, option (c) is obtained as the correct answer.

Note: The best way to approach this question is to figure out the form of the required equation. Looking at the options, the form of the parabola can be obtained easily. One way to figure out the answer would be to check the latus rectum. Since the length of the latus rectum is available from the question, the answer can be computed easily in less time.

Complete step-by-step answer:

The vertex and the latus rectum of a parabola are given as $\left( a,b \right)$ and \[l\] respectively in the question.

Since the axis of the parabola is along the positive direction of the $y-$axis, we can figure out that the form of the required parabola would be \[{{x}^{2}}=4ay\]. Latus rectum is indicated by LR and the given point of the vertex is termed as A. We can represent the details as shown in the figure below.

The vertex, A is $\left( a,b \right)$, so we can write the equation for the parabola as,

\[{{\left( x-a \right)}^{2}}=4c\left( y-b \right)\ldots \ldots \ldots (i)\]

Since the coordinate of the vertex is \[a\], the term $c$ has been used in the equation above.

From the figure, we can see that the latus rectum is perpendicular to the axis of the parabola and is represented in the figure as LR. We know that the length of the latus rectum for the form of parabola, \[{{x}^{2}}=4ay\] is \[4c\]. Also, it is already given to us in the question as \[l\].

Therefore, we can relate the data and we can write the term \[4c=l\].

After substituting this relation in equation \[(i)\], we get the equation of the parabola as,

\[{{\left( x-a \right)}^{2}}=l\left( y-b \right)\]

Hence, option (c) is obtained as the correct answer.

Note: The best way to approach this question is to figure out the form of the required equation. Looking at the options, the form of the parabola can be obtained easily. One way to figure out the answer would be to check the latus rectum. Since the length of the latus rectum is available from the question, the answer can be computed easily in less time.

Recently Updated Pages

Write a composition in approximately 450 500 words class 10 english JEE_Main

Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main

What is the common property of the oxides CONO and class 10 chemistry JEE_Main

What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main

If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main

The area of square inscribed in a circle of diameter class 10 maths JEE_Main