Step 1: Understanding the Concept:
We are given a geometric property involving the normal to a curve. We need to formulate a differential equation based on this property and solve it to find the family of curves.
Step 2: Key Formula or Approach:
The equation of the normal at a point $P(x, y)$ to the curve $y = f(x)$ is $Y - y = -\frac{dx}{dy}(X - x)$.
Find the coordinates of $Q$ where the normal intersects the X-axis (set $Y=0$).
Use the distance formula for $l(PQ)$ and set it equal to $k$ to form a differential equation.
Step 3: Detailed Explanation:
The equation of the normal at $P(x, y)$ is:
\[ Y - y = -\frac{1}{y'} (X - x) = -\frac{dx}{dy} (X - x) \]
It meets the X-axis at $Q(X_0, 0)$. Substitute $Y = 0$:
\[ -y = -\frac{dx}{dy} (X_0 - x) \]
\[ y \frac{dy}{dx} = X_0 - x \implies X_0 = x + y \frac{dy}{dx} \]
So, the coordinates of $Q$ are $(x + y \frac{dy}{dx}, 0)$.
The distance $l(PQ)$ is given by the distance formula:
\[ l(PQ) = \sqrt{(x + y \frac{dy}{dx} - x)^2 + (0 - y)^2} \]
\[ l(PQ) = \sqrt{\left(y \frac{dy}{dx}\right)^2 + y^2} = |y| \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \]
We are given that $l(PQ) = k$. Squaring both sides:
\[ y^2 \left[ 1 + \left(\frac{dy}{dx}\right)^2 \right] = k^2 \]
\[ y^2 + y^2 \left(\frac{dy}{dx}\right)^2 = k^2 \]
\[ y^2 \left(\frac{dy}{dx}\right)^2 = k^2 - y^2 \]
Taking the square root:
\[ y \frac{dy}{dx} = \pm \sqrt{k^2 - y^2} \]
Separate variables:
\[ \frac{y}{\sqrt{k^2 - y^2}} dy = \pm dx \]
Integrate both sides:
\[ \int \frac{y}{\sqrt{k^2 - y^2}} dy = \int \pm 1 dx \]
Let $t = k^2 - y^2$, then $dt = -2y dy$, so $y dy = -\frac{dt}{2}$.
\[ \int -\frac{1}{2\sqrt{t}} dt = \pm x + C \]
\[ -\frac{1}{2} (2\sqrt{t}) = \pm x + C \]
\[ -\sqrt{k^2 - y^2} = \pm x + C \]
We are given that the curve passes through the point $(0, k)$. Substitute $x=0, y=k$:
\[ -\sqrt{k^2 - k^2} = \pm 0 + C \implies 0 = C \]
So the equation simplifies to:
\[ -\sqrt{k^2 - y^2} = \pm x \]
Squaring both sides to eliminate the sign and the radical:
\[ k^2 - y^2 = x^2 \]
\[ x^2 + y^2 = k^2 \]
Step 4: Final Answer:
The equation of the curve is $x^2 + y^2 = k^2$.