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The general equation to a system of parallel chords in the parabola ${{y}^{2}}=\dfrac{25}{7}x$ is \[4x-y+k=0\]. What is the equation to the corresponding diameter?

Answer Verified Verified
Hint: The diameter of a parabola is defined as a line bisecting the system of parallel chords of a parabola. For ${{y}^{2}}=4ax$, it is given by \[y=\dfrac{2a}{m}\] where \[m\] is the slope of the chords.

Complete step-by-step answer:
It is given in the question that the general equation to a system of parallel chords in the parabola ${{y}^{2}}=\dfrac{25}{7}x$ is \[4x-y+k=0\].
We know that the general equation of a straight line is given by $y=mx+c$, where the term \[m\]represents the slope of the line and the term \[c\] represents the intercept.
We can convert the equation \[4x-y+k=0\] to the general form by rearranging the terms as below,
\[y=4x+k\]
When we compare the two equations, we get that the slope as \[m=4\].
We know that the general equation of the parabola is given by ${{y}^{2}}=4ax$. Now, let us consider the equation of the parabola given in the question, ${{y}^{2}}=\dfrac{25}{7}x$.
Now, let us convert the given equation into the general form. For that, we have to multiply and divide the RHS by \[4\]. This can be done as shown below,
$\begin{align}
  & {{y}^{2}}=\dfrac{4}{4}\times \dfrac{25}{7}x \\
 & {{y}^{2}}=4\times \left( \dfrac{25}{4\times 7} \right)x \\
 & {{y}^{2}}=4\times \left( \dfrac{25}{28} \right)x \\
\end{align}$
On comparing the above equation of the parabola with the general form, we get that $a=\dfrac{25}{28}$.
We know that the equation of diameter of the parabola is given by \[y=\dfrac{2a}{m}\]. Therefore, we can substitute the values of \[m=4\] and $a=\dfrac{25}{28}$ in it to get the diameter.
So, we get the diameter as
\[\begin{align}
  & y=\dfrac{2\times \dfrac{25}{28}}{4} \\
 & y=\dfrac{\dfrac{25}{14}}{4} \\
 & y=\dfrac{25}{56} \\
\end{align}\]
Therefore, the diameter of the parabola obtained is \[y=\dfrac{25}{56}\].
Note: Diameter of a parabola is also defined as the locus of all the mid-points of a system of parallel chords of a parabola. So, there is an alternate method to solve this question. From the equation of chord, $x$ can be substituted in the equation of parabola to formulate a quadratic equation. The roots of which will be the ordinates of point of intersection. The locus of midpoint of these ordinates would give the diameter.
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