Question

The differential equation of the circles having their centres on the line $y = 8$ and touching the $x$-axis is

• A. $(y - 8)^2\left[1 - \left(\frac{dy}{dx}\right)^2\right] = 64$
• B. $(y - 8)^2\left[1 + \left(\frac{dy}{dx}\right)^2\right] = 64$
• C. $(y - 8)\left[1 + \left(\frac{dy}{dx}\right)^2\right] = 64$
• D. $y^2\left(1 + \frac{dy}{dx}\right) = 64$

Hint:

For circles touching an axis, the radius equals the perpendicular distance of the centre from that axis.

Answer 1

The Correct Option is B

Step 1: Writing the general equation of the circle.
Since the centre lies on the line $y = 8$, let the centre be $(h, 8)$. As the circle touches the $x$-axis, its radius is $8$. \[ (x - h)^2 + (y - 8)^2 = 64 \]
Step 2: Differentiating with respect to $x$.
\[ 2(x - h) + 2(y - 8)\frac{dy}{dx} = 0 \] \[ x - h = -(y - 8)\frac{dy}{dx} \]
Step 3: Eliminating the parameter $h$.
Substitute $(x - h)$ into the original equation: \[ (y - 8)^2\left[1 + \left(\frac{dy}{dx}\right)^2\right] = 64 \]
Step 4: Conclusion.
The required differential equation is \[ (y - 8)^2\left[1 + \left(\frac{dy}{dx}\right)^2\right] = 64 \]