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Step-by-Step Solution
Step 1: Understand the given elements
We have a parabola given by the equation
$x^2 = 4y$.
A fixed point is
$(0, -1)$.
We consider a variable point
$(x_1, y_1)$
on the parabola and then find the locus of a point that divides the line segment between
$(0, -1)$
and
$(x_1, y_1)$
in the ratio
$1:2$
internally.
Step 2: Parametrize the point on the parabola
For the parabola
$x^2 = 4y$,
a convenient parameterization is
$(x_1, y_1) = (2t, t^2)$,
where $t$ is a real parameter.
Step 3: Apply the section formula
Let
$(X, Y)$
be the point dividing the segment from
$A(0, -1)$
to
$B(2t, t^2)$
in the ratio
$1:2$
internally. By the section formula, if a point divides a segment
$AB$
in the ratio
$m:n$
(with
$A$
corresponding to
$n$
and
$B$
to
$m$),
its coordinates are
$
\left(
\frac{m x_A + n x_B}{m + n},\,
\frac{m y_A + n y_B}{m + n}
\right).
$
Here, $m=1$ and $n=2$, giving:
$
(X, Y) = \left(
\frac{1 \cdot 2t + 2 \cdot 0}{1 + 2},\,
\frac{1 \cdot t^2 + 2 \cdot (-1)}{1 + 2}
\right)
=
\left(
\frac{2t}{3},\,
\frac{t^2 - 2}{3}
\right).
$
Step 4: Express the locus in terms of $X$ and $Y$
From
$X = \frac{2t}{3}$,
we get
$t = \frac{3X}{2}$.
Also, from
$Y = \frac{t^2 - 2}{3}$,
we get
$3Y = t^2 - 2,$
so
$t^2 = 3Y + 2$.
Step 5: Eliminate the parameter $t$
Since
$t = \frac{3X}{2}$,
$t^2 = \left(\frac{3X}{2}\right)^2 = \frac{9X^2}{4}$.
Equating the two expressions for
$t^2$,
we get:
$
\frac{9X^2}{4} = 3Y + 2.
$
Multiply both sides by
$4$:
$
9X^2 = 12Y + 8.
$
Rearrange to get the locus in standard form:
$
9X^2 - 12Y = 8.
$
Step 6: Conclude the locus equation
Therefore, the locus of the point that divides the line segment joining
$(0, -1)$
and
$(2t, t^2)$
in the ratio
$1:2$
is
$
9x^2 - 12y = 8,
$
which matches the correct option.
