Question

The equation of the parabola with vertex at origin, which passes through the point $( - 3,7)$ and axis along the $x - axis$ is
A) ${y^2} = 49x$
B) $3{y^2} = - 49x$
C) $3{y^2} = 49x$
D) ${x^2} = - 49$

Answer 1

Parabola is a U-shaped plane curved where any point is at an equal distance from the fixed point known as the focus and from the fixed straight line which is known as the directrix. The standard equation of the parabola is ${(x - h)^2} = 4p(y - k)$
In this question, we need to determine the equation of the parabola whose vertex and a point are given so that we can substitute them in the standard equation to get the result.

Complete step-by-step solution:
The Parabola with the vertex at the Origin, which passes through the point $( - 3,7)$
Axis of the parabola given is $=$ $x - axis$, so, the equation of the parabola will be of the form of ${(y - k)^2} = 4a(x - h)$
According to the question, the vertex of the parabola is at the origin.
Therefore, $h = 0$ and $k = 0$
Put the values in the standard equation we get, ${y^2} = 4ax - - - - (i)$
Also, it is given that parabola passes through the point $( - 3,7)$
Therefore, substituting the values of $x = - 3$ and $y = 7$ in the equation (i) we get,
$\Rightarrow {y^2} = 4ax \\\ \Rightarrow ({7^2}) = 4a( - 3) \\\ \Rightarrow a = \dfrac{{ - 49}}{{12}} - - - - (ii) \\\$
Therefore, by equation (i) and (ii), the equation of the parabola is-
$\Rightarrow{y^2} = 4(\dfrac{{ - 49}}{{12}})x \\\ \Rightarrow 3{y^2} = - 49x \\\$
Hence, the equation of the parabola with vertex at origin, which passes through the point $( - 3,7)$ and axis along the $x - axis$ is $3{y^2} = - 49x$.

Hence the correct answer is option B.

Note: Always remember the given axis of the parabola i.e. x-axis or y-axis and use standard form accordingly. The parabola is symmetric about its axis and the axis is perpendicular to the directrix. The vertex always passes through the vertex and the focus. The tangent at vertex is parallel to the given directrix.