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The equation of the line joining the vertex of the parabola ${y^2} = 6x$ to the points on it whose abscissa is 24, is: a. $y \pm 2x = 0$b. $2y \pm x = 0$c. $x \pm 2y = 0$d. $2x \pm y = 0$

Last updated date: 25th Mar 2023
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Hint: If two points are known then find the equation of line using two point form. Abscissa is the x - coordinate of a point and ordinate is the y - coordinate of a point.

Given Equation of parabola is ${y^2} = 6x$

$\Rightarrow {y^2} = 6x{\text{ }}\left( 1 \right)$

Now, we have first find ordinate of points with abscissa 24 that lie on the given parabola

Let the ordinate be y

So, (24,y) should satisfy the given equation of parabola

putting value of point in equation 1 we get,

$\Rightarrow {y^2} = 144$

$\Rightarrow y = \pm 12$

So, there will be two points on the given equation with abscissa as 24. Let these points be

$\Rightarrow P = \left( {24,12} \right){\text{and }}Q = \left( {24, - 12} \right)$

So, vertex of the equation 1 will be

$\Rightarrow {\text{vertex }} = \left( {0,0} \right)$

So, equation of line joining vertex and point P will be,

Finding equation of line using two point form where points are vertex $\equiv \left( {0,0} \right) \equiv \left( {{x_1},{y_1}} \right){\text{ and P}} \equiv \left( {24,12} \right) \equiv \left( {{x_2},{y_2}} \right)$

$\Rightarrow \left( {y - {y_1}} \right) = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right) \Rightarrow y = \frac{x}{2} \Rightarrow 2y - x = 0{\text{ }}\left( 2 \right)$

Now, equation of line joining vertex and point Q will be,

Finding equation of line using two point form where points are vertex $\equiv \left( {0,0} \right) \equiv \left( {{x_1},{y_1}} \right){\text{ and Q}} \equiv \left( {24, - 12} \right) \equiv \left( {{x_2},{y_2}} \right)$

$\Rightarrow \left( {y - {y_1}} \right) = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right) \Rightarrow y = \frac{{ - x}}{2} \Rightarrow 2y + x = 0{\text{ }}\left( 3 \right)$

From equations 2 and 3 we get, $2y \pm x = 0$ is the equation of line required.

Correct option for the question will be (b).

Note: Understand the diagram properly whenever you are facing these kinds of problems and also never neglect signs otherwise you will get only one solution. A better knowledge of formulas will be an added advantage.