Let 'a' and 'b' be non-zero real numbers. Then, the equation `(ax^2+ by^2+c) (x^2-5xy+6y^2)` represents :

Options

A. four staright lines, when c = 0 and a, b are of the same sign
B. two straight lines and a circle, when a = b and c is of sign opposite to that of a
C. two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a
D. a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

Explanation

To solve the problem, we need to analyze the given equation:

\[ (ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0 \]

This equation can be interpreted as the product of two expressions being equal to zero. For the product to be zero, at least one of the factors must be zero.

### Step 1: Analyze the second factor

First, let's analyze the second factor:

\[ x^2 - 5xy + 6y^2 = 0 \]

This is a quadratic equation in terms of \(x\). We can factor it:

\[ x^2 - 5xy + 6y^2 = (x - 2y)(x - 3y) = 0 \]

From this factorization, we find two solutions:

\[ x - 2y = 0 \quad \text{or} \quad x - 3y = 0 \]

This gives us the lines:

\[ x = 2y \quad \text{and} \quad x = 3y \]

### Step 2: Analyze the first factor

Now, we need to analyze the first factor:

\[ ax^2 + by^2 + c = 0 \]

This is a conic section. The nature of this conic section depends on the signs of \(a\), \(b\), and \(c\).

- If \(a\) and \(b\) have the same sign and \(c = 0\), the equation represents a point at the origin.
- If \(a\) and \(b\) have the same sign and \(c \neq 0\), it represents an ellipse (or a circle if \(a = b\)).
- If \(a\) and \(b\) have opposite signs, it represents a hyperbola.

### Step 3: Combine the results

Now, we combine the results from both factors. The overall equation represents:

1. The pair of straight lines given by \(x = 2y\) and \(x = 3y\).
2. The conic section represented by \(ax^2 + by^2 + c = 0\).

Thus, the equation represents a pair of straight lines along with a conic section (which could be a circle, ellipse, or hyperbola depending on the values of \(a\), \(b\), and \(c\)).

### Conclusion

Therefore, the equation \((ax^2 + by^2 + c)(x^2 - 5xy + 6y^2) = 0\) represents a pair of straight lines and a conic section.