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For what point of the parabola the ${y^2} = 18x$ ordinate equal to three times the abscissa?

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Last updated date: 23rd Jul 2024
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Answer
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Hint: Substitute the relation between abscissa and ordinate into the given parabola equation.


Complete step by step answer:

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As, we know that we are given with the parabola

${y^2} = 18x$ …………………………………………………….. (1)

We know that abscissa is the x-coordinate and ordinate is the y-coordinate.

Let the abscissa and ordinate of the point on parabola be \[{x_1}\] and \[{y_1}\] respectively.

So, according to the given condition. Ordinate of the point will be equal to three times the abscissa i.e. \[{y_1} = 3{x_1}\].

So, as the points lie on the parabola the coordinates of the points must satisfy the equation of parabola.

Now substituting the coordinates of the points in equation (1) we get,

$

   \Rightarrow {y_1}^2 = 18{x_1} \\

  \Rightarrow {\left( {3{x_1}} \right)^2} = 18{x_1} \\

   \Rightarrow {9x_1}\left( {{x_1} - 2} \right) = 0 \\

   \Rightarrow {x_1} = 0,2 \\

$

Hence,

For ${x_1} = 0 \Rightarrow {y_1} = 0$, Point$\left( {0,0} \right)$

For ${x_1} = 2 \Rightarrow {y_1} = 6$, Point$\left( {2,6} \right)$

 

Note: Whenever we come across these types of problems then we have to remember that if a point lies on the curve then its coordinates must satisfy the equation of the curve.