If O is the vertex of the of the parabola x² = 4y, Q is the point on parabola. If C is the locus of point which divides OQ in ratio 2:3. The equation of chord of C which bisected at point (1, 2).

Let us break down the solution to find the equation of the chord step-by-step.

Step 1: Understand the given parabola and points
The given parabola is:
$x^2=4y$
The vertex of this parabola is at the origin, so $O=(0,0)$.
Let $Q$ be a general point on the parabola. We can write the coordinates of $Q$ in parametric form as:
$Q=(2t,t^2)$
since $(2t{)}^2=4(t^2)$ satisfies the equation of the parabola $x^2=4y$.

Step 2: Find the locus of the point dividing OQ in the ratio 2:3
Let $P(h,k)$ be the point that divides the line segment joining $O(0,0)$ and $Q(2t,t^2)$ internally in the ratio $2:3$.
Using the section formula, the coordinates of $P(h,k)$ are:
$h=\frac{2\left(2t\right)+3\left(0\right)}{2+3}=\frac{4t}{5}$
$k=\frac{2\left(t^2\right)+3\left(0\right)}{2+3}=\frac{2t^2}{5}$
From the equation for $h$, we get:
$t=\frac{5h}{4}$
Substitute this value of $t$ into the equation for $k$:
$k=\frac{2}{5}{\left(\frac{5h}{4}\right)}^2=\frac{2}{5}\left(\frac{25h^2}{16}\right)=\frac{5h^2}{8}$
Simplifying this relation:
$8k=5h^2\Rightarrow h^2=\frac{8}{5}k$
Replacing $h$ with $x$ and $k$ with $y$, the equation of the locus curve $C$ is:
$x^2=\frac{8}{5}y\Rightarrow 5x^2-8y=0$

Step 3: Equation of the chord of C bisected at (1, 2)
The equation of a chord of a conic section bisected at a given point $(x_1,y_1)$ is given by:
$T=S_1$
Here, the equation of the curve is $S=5x^2-8y=0$, and the point is $(x_1,y_1)=(1,2)$.
Let us find $T$ and $S_1$:

$T=5x{x}_1-4(y+y_1)=5x\left(1\right)-4(y+2)=5x-4y-8$
$S_1=5x_1^2-8y_1=5(1{)}^2-8(2)=5-16=-11$
Now, set $T=S_1$:
$5x-4y-8=-11$
$5x-4y-8+11=0$
$5x-4y+3=0$

Therefore, the equation of the chord of C bisected at (1, 2) is $5x-4y+3=0$.