© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the Vertex and Focus
Given that the parabola's axis lies along the x-axis, and its vertex and focus lie on the positive x-axis.
• Vertex is at distance 2 from the origin on the positive x-axis → Vertex at $(2, 0)$.
• Focus is at distance 4 from the origin on the positive x-axis → Focus at $(4, 0)$.
Step 2: Determine the Parameter $p$
For a horizontal parabola of the form
$ (y - k)^2 = 4p (x - h) $,
the vertex is $(h, k)$ and the focus is $(h + p, k)$.
Here, $h = 2$, $k = 0$, and the focus $(4,0)$ gives $h + p = 4$.
Hence, $p = 2$.
Step 3: Write the Equation of the Parabola
Substituting $h=2$, $k=0$, and $p=2$ into the standard form:
$ (y - 0)^2 = 4 \times 2 \,(x - 2) $
So,
$ y^2 = 8(x - 2) $
or
$ y^2 = 8x -16 $.
Step 4: Check Each Given Point
(a) Checking (5, $2\sqrt{6}$)
Compute $y^2$ and compare with the right-hand side:
$ y^2 = (2\sqrt{6})^2 = 4 \times 6 = 24$.
Right-hand side: $8(5 - 2) = 8 \times 3 = 24$.
Both sides match → (5, $2\sqrt{6}$) lies on the parabola.
(b) Checking (6, $4\sqrt{2}$)
Compute $y^2$ and compare:
$ y^2 = (4\sqrt{2})^2 = 16 \times 2 = 32$.
Right-hand side: $8(6 - 2) = 8 \times 4 = 32$.
Both sides match → (6, $4\sqrt{2}$) lies on the parabola.
(c) Checking (8, 6)
Compute $y^2$ and compare:
$ y^2 = 6^2 = 36$.
Right-hand side: $8(8 - 2) = 8 \times 6 = 48$.
36 ≠ 48 → (8, 6) does not lie on the parabola.
(d) Checking (4, –4)
Compute $y^2$ and compare:
$ y^2 = (-4)^2 = 16$.
Right-hand side: $8(4 - 2) = 8 \times 2 = 16$.
Both sides match → (4, –4) lies on the parabola.
Step 5: Conclusion
Based on the calculations, the point that does not satisfy the parabola's equation is $(8, 6)$.
Thus, $(8, 6)$ does not lie on the parabola.
