Question

The coordinates of the focus of the parabola described parametrically by $x=5t^2+2,y=10t+4$ are:

• A. $(7,4)$
• B. $(3,4)$
• C. $(3,-4)$
• D. $(-7,4)$

Answer 1

The correct option is A

$(7,4)$

Explanation for the correct option:

Given that : $$\begin{gathered} x=5t^2+2, \\ y=10t+4 \end{gathered}$$

Now solving it,

$$\begin{gathered} x=5t^2+2 \\ \Rightarrow \left(x-2\right)=5t^2 \\ \Rightarrow t^2=\frac{\left(x-2\right)}{5}...\left(i\right) \end{gathered}$$

Now, $$\begin{gathered} y=10t+4 \\ \Rightarrow t=\frac{\left(y-4\right)}{10}....\left(ii\right) \end{gathered}$$

Squaring equation $\left(ii\right)$ and equating with $\left(i\right)$

$$\begin{gathered} \Rightarrow \frac{{\left(y-4\right)}^2}{100}=\frac{\left(x-2\right)}{5} \\ \Rightarrow {\left(y-4\right)}^2=20\left(x-2\right) \end{gathered}$$

Let $X=x-2\&Y=y-4$

$$\begin{gathered} \therefore Y^2=20X \\ \Rightarrow Y^2=4\left(5\right)X \end{gathered}$$

Therefore, coordinate of focus is, $(5,0)$

As we know that if the equation of parabola is $y^2=4ax$then the coordinate of the focus is $(a,0)$

Now

$$\begin{gathered} x-2=5\Rightarrow x=7 \\ y-4=0\Rightarrow y=4 \end{gathered}$$

Hence the required coordinates are $(7,4)$

Hence, option (A) is the correct answer.