The coordinates of the focus of the parabola | vedclass

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(A) Given parametric equations are $x = 5t^2 + 2$ and $y = 10t + 4$.From the second equation, $10t = y - 4$, so $t = \frac{y - 4}{10}$.Substituting th

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$(7, 4)$

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(A) Given parametric equations are $x = 5t^2 + 2$ and $y = 10t + 4$.From the second equation, $10t = y - 4$, so $t = \frac{y - 4}{10}$.Substituting this into the first equation: $x = 5\left(\frac{y - 4}{10}\right)^2 + 2$.$x - 2 = 5 \cdot \frac{(y - 4)^2}{100} = \frac{(y - 4)^2}{20}$.Thus, $(y - 4)^2 = 20(x - 2)$.This is of the form $(y - k)^2 = 4a(x - h)$, where $(h, k) = (2, 4)$ is the vertex and $4a = 20$, so $a = 5$.The focus of this parabola is $(h + a, k) = (2 + 5, 4) = (7, 4)$.

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

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