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 Question:
A circle with center on \( y = 5 \) has the form \( (h, 5) \). Since it touches the X-axis, its radius must be equal to the distance from center to X-axis, which is 5.
Step 2: Detailed Explanation:
Equation: \( (x - h)^2 + (y - 5)^2 = 5^2 \).
\( (x - h)^2 + (y - 5)^2 = 25 \).
Differentiating with respect to \( x \):
\( 2(x - h) + 2(y - 5)\frac{dy}{dx} = 0 \).
\( x - h = -(y - 5) \frac{dy}{dx} \).
Substituting \( (x - h) \) back into the original equation:
\( [-(y - 5) \frac{dy}{dx}]^2 + (y - 5)^2 = 25 \).
\( (y - 5)^2 \left(\frac{dy}{dx}\right)^2 + y^2 - 10y + 25 = 25 \).
\( (5 - y)^2 \left(\frac{dy}{dx}\right)^2 + y^2 - 10y = 0 \).
Step 3: Final Answer:
The differential equation is \( (5 - y)^2 \left(\frac{dy}{dx}\right)^2 + y^2 - 10y = 0 \).