Question

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

Answer 1

Hint: Here we will use the given relation between the abscissa and ordinate and then substitute it in the given equation of parabola.

Complete step-by-step answer:

The given equation of the parabola is:-

\[{y^2} = 18x\]…………………………………(1)

Now we know that ordinate is the other name for y coordinate and abscissa is the other name for x coordinate.

Let the x coordinate and the y coordinate of the point on the parabola be \[{x_1}\]and \[{y_1}\] respectively

Now it is given that ordinate or y coordinate is equal to three times of the abscissa or x coordinate

Hence,

\[{y_1} = 3{x_1}\]……………………………..(2)

Now substituting the coordinates of the point on parabola in equation 1 we get:-

\[{\left( {{y_1}} \right)^2} = 18{\left( {{x_1}} \right)^2}\]

Solving it further we get:-

\[{y_1}^2 = 18{x_1}\]

Now substituting the value \[{y_1}\] from equation (2) in above equation we get:

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

Simplifying it further we get:-

\[9{x_1}^2 - 18{x_1} = 0\]

Taking \[9{x_1}\] as common we get:-

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

Solving for \[{x_1}\] we get:-

\[{x_1} = \dfrac{0}{9};{x_1} - 2 = 0\]

\[ \Rightarrow {x_1} = 0;{x_1} = 2\]

Now substituting these values in equation 2 we get:-

When \[{x_1} = 0\]

\[{y_1} = 3\left( 0 \right)\]

\[ \Rightarrow {y_1} = 0\]

When \[{x_1} = 2\]

\[{y_1} = 3\left( 2 \right)\]

\[ \Rightarrow {y_1} = 6\]

Hence, required points are : - \[\left( {0,0} \right);\left( {2,6} \right)\]

Note: Students should take a note that when a point lies on the curve then it must satisfy its equation.

Also, students can check these points from the graph of the curve.

We can verify our answer by substituting the points in the equation of parabola.