Animated Solution for Mathematics - Conic Sections: The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^2=4y$, internally in the ratio 1: 2, is :

Imagine you are standing on the Cartesian plane, looking at the elegant curve of the parabola defined by the equation $x^2=4y$. It is a beautiful, symmetric shape opening upwards, with its vertex at the origin.

Now, imagine a fixed point $A$ sitting at $(0,-1)$, just below the origin. We are interested in a moving point $B$ that travels along this parabola. As $B$ moves, we want to track the path of a special point $P$ that divides the segment $AB$ in a fixed ratio of $1:2$.

Our Secret Weapon

To solve this, we need to describe the position of $B$ efficiently. Instead of juggling two variables and a constraint, we use the power of parametric coordinates.

For the parabola $x^2=4y$, we can define any point $B$ as $(2t,t^2)$. Here, the single parameter $t$ acts as a dial; as you turn $t$, $B$ slides smoothly along the parabola.

The Bridge

Consider the segment $AB$. Point $P(h,k)$ divides this segment internally in the ratio $1:2$. The Section Formula is our most reliable tool here, providing the weighted average of the coordinates of $A$ and $B$.

For the $x$-coordinate $h$, we have:

Substituting our known values $x_B=2t$ and $x_A=0$, we get:

We apply the same logic to the $y$-coordinate $k$:

With $y_B=t^2$ and $y_A=-1$, this becomes:

Eliminating the Parameter

We now have $h$ and $k$ expressed in terms of $t$. To find the locus, we must eliminate the parameter $t$. From our equation for $h$, we isolate $t$:

Now, we substitute this expression for $t$ into our equation for $k$:

Multiplying by $3$, we obtain:

Adding $2$ to both sides and multiplying by $4$, we arrive at:

Replacing $h$ and $k$ with the general variables $x$ and $y$, we obtain the final equation of the locus:

This is the equation of the locus. It is another parabola, shifted and scaled, representing the path traced by $P$.