Animated Solution for Mathematics - Conic Sections: Given the points $A(0,4)$ and $B(0,-4)$, the equation of the locus of the point $P(x,y)$ such that $|AP-BP|=6$ is .........

Imagine you are standing on a vast, flat coordinate plane. You have two fixed markers, $A$ at $(0,4)$ and $B$ at $(0,-4)$. These are your anchors, your points of reference.

Now, imagine a point $P(x,y)$ that is dancing across this plane, constrained by a specific rule: the absolute difference between its distance to $A$ and its distance to $B$ must always be exactly $6$. That is, $|AP-BP|=6$.

This is not just an algebraic exercise; it is the fundamental heartbeat of a hyperbola.

When we look at the condition $|AP-BP|=6$, we are looking at the very definition of a hyperbola. In the world of conic sections, while an ellipse is the locus of points where the sum of distances to two foci is constant, the hyperbola is the locus where the difference is constant.

Our points $A$ and $B$ are not just random coordinates; they are the foci of our hyperbola. Because they lie on the y-axis, we immediately know that our hyperbola will be vertically oriented, opening upwards and downwards.

To build the equation of this hyperbola, we need two key parameters: $a$ and $b$. The constant difference in the definition of a hyperbola is equal to the length of the transverse axis, which is $2a$.

Therefore, we can write $2a=6$, which gives us $a=3$. Squaring this, we get $a^2=9$. This is the first piece of our puzzle.

Next, we need to find the distance between the foci. The distance between $A(0,4)$ and $B(0,-4)$ is simply the difference in their y-coordinates, which is $8$. In the theory of hyperbolas, the distance between the foci is given by $2c$ (where $c=ae$). So, we set $2c=8$, which simplifies to $c=4$.

Now, we need the value of $b^2$. We have a powerful relationship in hyperbola geometry:

Substituting our known values, we have $4^2=3^2+b^2$. This simplifies to $16=9+b^2$.

Solving for $b^2$, we find:

We have everything we need. We know the hyperbola is vertical because the foci are on the y-axis. The standard equation for a vertical hyperbola is:

Plugging in our values for $a^2$ and $b^2$, we arrive at the final, elegant equation:

This equation is the mathematical signature of the path traced by point $P$. It is a testament to how coordinate geometry can transform a simple distance condition into a beautiful, structured curve.