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Question: If \( x = {t^2} + 2 \) and \( y = 2t \) represent the parametric equation of the parabola A. \( {x...

If x=t2+2x = {t^2} + 2 and y=2ty = 2t represent the parametric equation of the parabola
A. x2=4(y−2){x^2} = 4\left( {y - 2} \right)
B. 4x=(y−2)24x = {\left( {y - 2} \right)^2}
C. y2=4(x−2){y^2} = 4\left( {x - 2} \right)
D. (x−2)2=4y{\left( {x - 2} \right)^2} = 4y

Explanation

Solution

Hint : In order to determine the equation of the parabola from the given parametric equation , by finding the value of tt from the equation 2nd and substitute that value in the 1st equation .Simplify the equation to obtain the equation of parabola.

Complete step-by-step answer :
This is the question from the equation of parabola.
Here we are given that the equations x=t2+2x = {t^2} + 2 and y=2ty = 2t are the parametric equations of some parabola and we have to find the equation of the same parabola.
x=t2+2x = {t^2} + 2 ---(1)
y=2ty = 2t -----(2)
So, to find the equation of the parabola , we will be finding the value of tt from the equation (2) by dividing both sides of the equation by the number 22 , we get
y2=2t2 ⇒t=y2   \dfrac{y}{2} = \dfrac{{2t}}{2} \\\ \Rightarrow t = \dfrac{y}{2} \;
Now putting the above value of tt in the equation(1), the equation becomes
⇒x=(y2)2+2 x=y222+2 x=y24+2   \Rightarrow x = {\left( {\dfrac{y}{2}} \right)^2} + 2 \\\ x = \dfrac{{{y^2}}}{{{2^2}}} + 2 \\\ x = \dfrac{{{y^2}}}{4} + 2 \;
Transposing the constant term form the right-hand side to left-hand side , and then multiplying both sides of the equation with the number 4, we get

⇒x−2=y24 4(x−2)=y2 ⇒y2=4(x−2)  \Rightarrow x - 2 = \dfrac{{{y^2}}}{4} \\\ 4\left( {x - 2} \right) = {y^2} \\\ \Rightarrow {y^2} = 4\left( {x - 2} \right) \;

Therefore, the equation of parabola is y2=4(x−2){y^2} = 4\left( {x - 2} \right), option C is correct
So, the correct answer is “Option C”.

Note : Quadratic Equation: A quadratic equation is a equation which can be represented in the form of ax2+bx+ca{x^2} + bx + c where xx is the unknown variable and a,b,c are the numbers known where a≠0a \ne 0 .If a=0a = 0 then the equation will become linear equation and will no more quadratic .
Note:
1.Make sure the simplification of the equation is done correctly.
2. A parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
3. Graph of every quadratic equation is a parabola.