Question

The coordinates of the focus of the parabola described parametrically by \[x=5{{t}^{2}}+2\] and \[y=10t+4\] are
1) \[\left( 7,4 \right)\]
2) \[\left( 3,4 \right)\]
 3) \[\left( 3,-4 \right)\]
4) \[\left( -7,4 \right)\]

Answer 1

Hint: In this problem, we are given two parametric equations from which we have to find the coordinates of the parabola. We can first write the given equation in the squared term of t form. We can then make substitutions and solve for x and y to get the required coordinates value.

Complete step by step solution:
Here we are given two parametric equations from which we have to find the coordinates of the parabola.
We know that the given parametric equation is,
\[x=5{{t}^{2}}+2\]
We can now write the above given equation as,
\[\Rightarrow \dfrac{x-2}{5}={{t}^{2}}\] ………. (1)
We are given another parametric equation,
\[y=10t+4\]
We can now write the above given equation as,
\[\Rightarrow \dfrac{y-4}{10}=t\] ……….. (2)
We can now substitute (2) in (1), we get
\[\Rightarrow \dfrac{x-2}{5}={{\left( \dfrac{y-4}{10} \right)}^{2}}\]
We can now simplify the above step, we get
\[\begin{align}
  & \Rightarrow \dfrac{x-2}{5}=\dfrac{{{\left( y-4 \right)}^{2}}}{100} \\
 & \Rightarrow 20\left( x-2 \right)={{\left( y-4 \right)}^{2}} \\
\end{align}\]
We can now write the above step in the form,
\[\Rightarrow 20X={{Y}^{2}}\]
Where, \[X=x-2\] and \[Y=y-4\].
We know that the coordinates of the focus is \[\left( 5,0 \right)\].
\[\begin{align}
  & \Rightarrow 5=x-2,0=y-4 \\
 & \Rightarrow x=7,y=4 \\
\end{align}\]

Therefore, the coordinates of the parabola is option 1) \[\left( 7,4 \right)\].

Note: We should always remember that the parametric form of the equation can be converted into its original form by substitution method. We should know how to draw diagrams and collect the required data from it. We should also substitute the required value to get the exact answer for the given question.