Question

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

Answer 1

Hint: In order to determine the equation of the parabola from the given parametric equation , by finding the value of $ t $ 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 = {t^2} + 2 $ and $ y = 2t $ are the parametric equations of some parabola and we have to find the equation of the same parabola.
 $ x = {t^2} + 2 $ ---(1)
 $ y = 2t $ -----(2)
So, to find the equation of the parabola , we will be finding the value of $ t $ from the equation (2) by dividing both sides of the equation by the number $ 2 $ , we get
 $
  \dfrac{y}{2} = \dfrac{{2t}}{2} \\
\Rightarrow t = \dfrac{y}{2} \;
  $
Now putting the above value of $ t $ in the equation(1), the equation becomes
 $
\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
\[
\Rightarrow x - 2 = \dfrac{{{y^2}}}{4} \\
  4\left( {x - 2} \right) = {y^2} \\
\Rightarrow {y^2} = 4\left( {x - 2} \right) \;
 \]
Therefore, the equation of parabola is \[{y^2} = 4\left( {x - 2} \right)\], 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 $ a{x^2} + bx + c $ where $ x $ is the unknown variable and a,b,c are the numbers known where $ a \ne 0 $ .If $ a = 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.