$$\begin{gathered} \text{Let}O\text{be the vertex of the parabola}{y}^2=16x. \\ \end{gathered}$$

\text{The locus of centroid of}\Delta OPA\text{when}P\text{lies on parabola, and}A\text{lies on}x-\mathrm{axis}\text{and}\angle OPA=90°\text{is:}

$$\begin{gathered} \\ \end{gathered}$$

Correct Answer: The correct locus equation is 9𝑦² = 8(3𝑥 − 16) (which corresponds to Option 2).

Let us solve the problem step-by-step to find the locus of the centroid of ΔOPA.

Step 1: Parametric coordinates of point P on the parabola
The equation of the parabola is:

y^2=16x

Comparing this with the standard parabola y^2=4ax, we find 4a=16\Rightarrow a=4.
The vertex O is at the origin, i.e., O=(0,0).
Any point P on the parabola can be represented in parametric form as:

P=(a{t}^2,2at)=(4t^2,8t)

Step 2: Finding coordinates of point A on the x-axis
Point A lies on the x-axis, so its coordinates are A=(x_1,0).
We are given that \angle OPA=90°, which means the line segment OP is perpendicular to line segment PA. Therefore, the product of their slopes must equal -1:

m_{OP}·m_{PA}=-1

The slope of OP is:

m_{OP}=\frac{8t-0}{4t^2-0}=\frac{2}{t}

The slope of PA is:

m_{PA}=\frac{0-8t}{x_1-4t^2}

Setting m_{OP}·m_{PA}=-1:

\left(\frac{2}{t}\right)·\left(\frac{-8t}{x_1-4t^2}\right)=-1

Simplifying the expression:

\frac{-16}{x_1-4t^2}=-1\Rightarrow x_1-4t^2=16\Rightarrow x_1=4t^2+16

Thus, the coordinates of point A are (4t^2+16,0).

Step 3: Centroid of ΔOPA
Let G(h,k) be the centroid of triangle OPA. The vertices are O(0,0), P(4t^2,8t), and A(4t^2+16,0).
Using the formula for centroid:

h=\frac{0+4t^2+(4t^2+16)}{3}=\frac{8t^2+16}{3}

k=\frac{0+8t+0}{3}=\frac{8t}{3}

Step 4: Eliminating parameter t to find the locus
From the equation for k:

t=\frac{3k}{8}

Substitute this value of t into the equation for h:

h=\frac{8{\left(\frac{3k}{8}\right)}^2+16}{3}

3h=8·\frac{9k^2}{64}+16

3h=\frac{9k^2}{8}+16

Multiply both sides by 8:

24h=9k^2+128

Rearranging terms:

9k^2=24h-128=8(3h-16)

Replacing (h,k) with (x,y) gives the equation of locus:

9y^2=8(3x-16)