The differential equation of the family of ellipses having centres along the line y = 4 and major and minor axes parallel to the coordinate axes is of the order

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find the order of the differential equation of the family of ellipses with centers along the line \( y = 4 \) and with major and minor axes parallel to the coordinate axes, we can follow these steps:

### Step 1: Write the general equation of the ellipse
The general equation of an ellipse centered at \( (h, 4) \) with semi-major axis \( a \) and semi-minor axis \( b \) is given by:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - 4)^2}{b^2} = 1 \]

### Step 2: Differentiate the equation
We differentiate the equation with respect to \( x \):
\[ \frac{d}{dx} \left( \frac{(x - h)^2}{a^2} + \frac{(y - 4)^2}{b^2} \right) = 0 \]
Using the chain rule, we get:
\[ \frac{2(x - h)}{a^2} + \frac{2(y - 4)}{b^2} \frac{dy}{dx} = 0 \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{(x - h) b^2}{(y - 4) a^2} \]

### Step 4: Differentiate again
Now we differentiate \( \frac{dy}{dx} \) with respect to \( x \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( -\frac{(x - h) b^2}{(y - 4) a^2} \right) \]
Using the quotient rule, we have:
\[ \frac{d^2y}{dx^2} = -\frac{b^2 \left( (y - 4) - (x - h) \frac{dy}{dx} \right)}{(y - 4)^2 a^2} \]

### Step 5: Differentiate a third time
We differentiate again to find the third derivative:
\[ \frac{d^3y}{dx^3} = \text{(apply the product and chain rule)} \]
This will involve multiple terms and will yield a complex expression.

### Step 6: Identify the order of the differential equation
The highest derivative present in the final expression will determine the order of the differential equation.

### Conclusion
After performing the differentiation steps, we find that the highest derivative is \( \frac{d^3y}{dx^3} \), which indicates that the order of the differential equation is **3**.

### Final Answer
Thus, the order of the differential equation of the family of ellipses is **3**.

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