Question

Axis of a parabola lies along $x$-axis. If its vertex and focus are at distances $2$ and $4$ respectively from the origin, on the positive $x$-axis then which of the following points does not lie on it ?

• A. $(4, -4)$
• B. $(5 , 2 \sqrt{6})$
• C. $(8, 6)$
• D. $ 6 , 4 \sqrt{2}$

Answer 1

The Correct Option is C

The given question involves identifying a point that does not lie on a parabola. The information provided indicates that the parabola's axis is along the \(x\)-axis with its vertex and focus located on the positive \(x\)-axis. Let's solve this step-by-step:

1. The vertex of the parabola is at a distance of \(2\) from the origin. Therefore, the vertex is at point \((2, 0)\).
2. The focus of the parabola is at a distance of \(4\) from the origin. Therefore, the focus is at point \((4, 0)\).
3. The standard equation of a parabola with vertex \((h, k)\) opening along the \(x\)-axis is given by:

\((y - k)^2 = 4a(x - h)\)

1. For this parabola, the distance between the vertex and the focus is \(2\) (from \((2, 0)\) to \((4, 0)\)), meaning \(a = 2\).
2. Substituting these values into the standard equation:

\((y - 0)^2 = 8(x - 2)\)

\(y^2 = 8(x - 2)\)

1. Now, we'll check each given option to see if it satisfies the equation \(y^2 = 8(x - 2)\):
   • \((4, -4)\):

\((-4)^2 = 8(4 - 2)\)

\(16 = 16\)

• This point lies on the parabola.
• \((5 , 2 \sqrt{6})\):

\((2 \sqrt{6})^2 = 8(5 - 2)\)

\(24 = 24\)

• This point lies on the parabola.
• \((8, 6)\):

\(6^2 = 8(8 - 2)\)

\(36 \neq 48\)

• This point does not lie on the parabola.
• \((6, 4\sqrt{2})\):

\((4 \sqrt{2})^2 = 8(6 - 2)\)

\(32 = 32\)

• This point lies on the parabola.

1. Hence, the point \((8, 6)\) does not satisfy the equation of the parabola and does not lie on it.

Therefore, the correct answer is the point \((8, 6)\).