Parabola Locus Derivation
The given parabola is $x^2 = 4y$, with vertex O at (0, 0). A point Q on the parabola can be represented parametrically as $Q(2t, t^2)$.
The point P divides the line segment OQ internally. For the conic C to be $5x^2=2y$, the internal division ratio $m:n$ must be $1:9$.
Using the section formula for P(h, k) dividing O(0,0) and Q(2t, t^2) in the ratio $1:9$:
$h = \frac{9(0) + 1(2t)}{1+9} = \frac{2t}{10} = \frac{t}{5}$
$k = \frac{9(0) + 1(t^2)}{1+9} = \frac{t^2}{10}$
From $h = \frac{t}{5}$, we get $t = 5h$. Substituting this into the expression for $k$:
$k = \frac{(5h)^2}{10} = \frac{25h^2}{10} = \frac{5h^2}{2}$
Therefore, the locus of P, which is the conic C, has the equation $5x^2 = 2y$. Let this equation be $S \equiv 5x^2 - 2y = 0$.
Chord Bisected Calculation
We need to find the equation of the chord of conic C ($5x^2 = 2y$) that is bisected at the point $(x_1, y_1) = (1, 2)$.
The standard equation for a chord bisected at $(x_1, y_1)$ is $T = S_1$, where:
- T is obtained by substituting $x^2 \to x x_1$ and $y \to \frac{y+y_1}{2}$ in the conic's equation S: $T \equiv 5(x x_1) - 2(\frac{y+y_1}{2}) = 5x(1) - (y+2) = 5x - y - 2$
- S1 is the value of S when $(x_1, y_1)$ is substituted: $S_1 \equiv 5x_1^2 - 2y_1 = 5(1)^2 - 2(2) = 5 - 4 = 1$
Equating $T = S_1$ gives the chord equation:
$5x - y - 2 = 1$
$5x - y - 3 = 0$
This equation corresponds to Option C.
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