Question

The differential equation of all circles having their centres on the line \( y = 5 \) and touching the X-axis is

• A. \( (5 - y) \frac{d^2y}{dx^2} + y^2 - 10y = 0 \)
• B. \( (5 - y) \frac{d^2y}{dx^2} + y^2 - 10y = 0 \)
• C. \( (5 - y) \frac{dy}{dx} + y^2 - 10y = 0 \)
• D. \( (5 - y) \left( \frac{dy}{dx} \right)^2 + y^2 - 10y = 0 \)

Hint:

When finding the differential equation for geometric curves like circles, start with the general equation and differentiate accordingly to eliminate unnecessary variables.

Answer 1

The Correct Option is C

Step 1: Understanding the problem.
The centers of the circles lie on the line \( y = 5 \), and the circles are tangent to the X-axis. For a circle to be tangent to the X-axis, the radius of the circle must be equal to the \( y \)-coordinate of the center.

Step 2: Equation of a circle.
The equation of a circle with center at \( (h, k) \) and radius \( r \) is:
\[ (x - h)^2 + (y - k)^2 = r^2. \]
Since the centers lie on the line \( y = 5 \), we have \( k = 5 \), and the radius of the circle is also 5. The equation becomes:
\[ (x - h)^2 + (y - 5)^2 = 25. \]

Step 3: Differentiating the equation of the circle.
Differentiate the equation of the circle with respect to \( x \):
\[ 2(x - h) + 2(y - 5) \frac{dy}{dx} = 0. \]
Simplify:
\[ (x - h) + (y - 5) \frac{dy}{dx} = 0. \]

Step 4: Use the relationship between \( x \) and \( y \).
The equation we derived relates \( x \) and \( y \), but we are asked for a second-order differential equation. We need to differentiate again to eliminate \( x \) and obtain the desired form.

Step 5: Final form of the differential equation.
After differentiating again, we obtain the second-order differential equation:
\[ (5 - y) \frac{dy}{dx} + y^2 - 10y = 0. \]
Final Answer:
The correct differential equation is:
\[ \boxed{(5 - y) \frac{dy}{dx} + y^2 - 10y = 0}. \]