Q.A rod $AB$ of length $15$ cm rests in between two coordinate axes in such a way that the end point $A$ lies on $x$-axis and end point $B$ lies on $y$-axis. A point $P(x, y)$ is taken on the rod in such a way that $AP = 6$ cm. Show that the locus of $P$ is an ellipse.

Concept: Locus of Point — We find the relation between $x$ and $y$ that $P$ must satisfy as the rod slides.

Let $A = (a, 0)$ on the $x$-axis and $B = (0, b)$ on the $y$-axis.

Since $AB = 15$, we have $a^2 + b^2 = 15^2 = 225$.

Point $P(x, y)$ divides $AB$ such that $AP = 6$, so $PB = 15 - 6 = 9$.

Using the section formula (internal division in ratio $AP:PB = 6:9 = 2:3$):

Thus $a = \frac{5x}{3}$ and $b = \frac{5y}{2}$. Substitute into $a^2 + b^2 = 225$: …

The key insight is that the rod’s ends are constrained to the axes, but the rod itself can take any orientation. Point $P$ is fixed relative to the rod — it is 6 cm from $A$ and therefore 9 cm from $B$. So $P$ divides $AB$ in the ratio $AP:PB = 6:9 = 2:3$. This ratio stays constant no matter how the rod slides.

Because $A$ and $B$ always lie on the $x$-axis and $y$-axis respectively, we can write their coordinates as $A(a,0)$ and $B(0,b)$. The distance $AB$ is fixed at 15, giving $a^2 + b^2 = 225$. The point $P$ is the point that divides $AB$ internally in the ratio $2:3$. Using the section formula, we express $x$ and $y$ in terms of $a$ and $b$, then eliminate $a$ and $b$ using the distance constraint.

Let’s work through it.

1. Set up coordinates. Let $A = (a, 0)$ on the $x$-axis and $B = (0, b)$ on the $y$-axis, where $a \ge 0$, $b \ge 0$ (the rod is in the first quadrant). The length condition gives:

So:

2. Locate point $P$ using the section formula. $P$ divides $AB$ such that $AP = 6$ and $PB = 9$, so $AP:PB = 2:3$. For internal division, the coordinates of $P$ are:

Hence:

3. Substitute into the length constraint. From $a^2 + b^2 = 225$:

4. Simplify to standard form. Divide through by 25:

Multiply through by 36 (LCM of 9 and 4) to clear denominators:

Or, in standard ellipse form: …