Understanding the Parabola

The axis of the parabola lies along the x-axis, and both its vertex and focus are on the positive x-axis. The vertex is at a distance of 2 units from the origin, and the focus is at a distance of 4 units from the origin. Therefore, the vertex V is at (2, 0) and the focus F is at (4, 0).

Step 1: Determine the Equation of the Parabola

Since the axis is horizontal, the standard equation of a parabola with vertex (h, k) is:

${(y-k)}^2=4a(x-h)$

Here, vertex V = (2, 0), so h = 2 and k = 0.

The distance from vertex to focus is |a|. Since the focus is at (4, 0) and vertex at (2, 0), the distance is 2 units. The parabola opens to the right (positive x-direction), so a is positive: a = 2.

Substitute h, k, and a into the equation:

$y^2=4\times 2\times (x-2)$

Simplify:

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

This is the equation of the parabola.

Step 2: Check Which Points Lie on the Parabola

For a point (x, y) to lie on the parabola, it must satisfy the equation $y^2=8(x-2)$.

Option 1: (5, 2√6)

Left side: $y^2={\left(2\sqrt{6}\right)}^2=4\times 6=24$

Right side: $8(5-2)=8\times 3=24$

24 = 24. The point lies on the parabola.

Option 2: (8, 6)

Left side: $y^2=6^2=36$

Right side: $8(8-2)=8\times 6=48$

36 ≠ 48. The point does NOT lie on the parabola.

Option 3: (6, 4√2)

Left side: $y^2={\left(4\sqrt{2}\right)}^2=16\times 2=32$

Right side: $8(6-2)=8\times 4=32$

32 = 32. The point lies on the parabola.

Option 4: (4, -4)

Left side: $y^2={(-4)}^2=16$

Right side: $8(4-2)=8\times 2=16$

16 = 16. The point lies on the parabola.

Final Answer

The point that does not lie on the parabola is (8, 6).

Related Topics

• Conic Sections
• Standard Equations of Parabolas
• Focus and Directrix of a Parabola
• Latus Rectum

Key Formulae and Theory

Standard Equation of a Parabola (Horizontal Axis):

Vertex at (h, k): ${(y-k)}^2=4a(x-h)$

Focus: (h + a, k)

Directrix: x = h - a

Latus Rectum Length: |4a|

Key Property: A parabola is the set of all points that are equidistant from the focus and the directrix.