The differential equation of all conics whose centre lie at the origin is of order

Options

A. 2
B. 3
C. 4
D. none of these

Explanation

To determine the order of the differential equation of all conics whose centers lie at the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the General Equation of Conics**:
The general equation of a conic section with its center at the origin can be expressed as:
\[ ax^2 + 2hxy + by^2 + c = 0 \]
where \(a\), \(h\), \(b\), and \(c\) are constants.

2. **Normalization of the Equation**:
To simplify the analysis, we can divide the entire equation by \(a\) (assuming \(a \neq 0\)):
\[ x^2 + \frac{2h}{a}xy + \frac{b}{a}y^2 + \frac{c}{a} = 0 \]
Let’s denote:
- \(C_1 = \frac{2h}{a}\)
- \(C_2 = \frac{b}{a}\)
- \(C_3 = \frac{c}{a}\)

Thus, we can rewrite the equation as:
\[ x^2 + C_1xy + C_2y^2 + C_3 = 0 \]

3. **Identifying the Number of Arbitrary Constants**:
The equation now contains three arbitrary constants \(C_1\), \(C_2\), and \(C_3\). These constants represent the different conics that can be formed.

4. **Relating Constants to the Order of the Differential Equation**:
The order of the differential equation is determined by the number of arbitrary constants present in the equation. Since we have three arbitrary constants, the order of the differential equation is:
\[ \text{Order} = 3 \]

5. **Conclusion**:
Therefore, the order of the differential equation of all conics whose centers lie at the origin is 3.

### Final Answer:
The order of the differential equation of all conics whose centers lie at the origin is **3**.

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