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The differential equation of the family ...

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

A

4

B

3

C

2

D

1

Text Solution

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The correct Answer is:
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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