Consider the following statements: $S_{1}$: The equation $ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$ represents a pair of straight lines. $S_{2}$: The equation $ax^{2}+2hxy+by^{2}=0$ always represents a pair of straight lines passing through the origin. Which of the following is correct?

Question

Accepted Answer

To justify why Option 1 is correct, let's analyze the given statements and options:

### Explanation of Statements:

1. **Statement $ S_1 $:** The equation $ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 $ represents a pair of straight lines if the determinant of the quadratic form is zero. This condition is given by $ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 $.

2. **Statement $ S_2 $:** The equation $ ax^2 + 2hxy + by^2 = 0 $ represents a pair of straight lines passing through the origin. This is always true because it is a homogeneous equation of degree 2, which can be factored into two linear factors, representing lines through the origin.

### Justification for Option 1:

- **If $ S_1 $ is true:** This means the equation $ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 $ represents a pair of straight lines. If $ c = 0 $, the equation reduces to $ ax^2 + 2hxy + by^2 + 2gx + 2fy = 0 $, which can be rewritten as $ ax^2 + 2hxy + by^2 = -(2gx + 2fy) $. Since $ ax^2 + 2hxy + by^2 = 0 $ is a homogeneous equation representing lines through the origin, $ S_2 $ is always true when $ S_1 $ is true and $ c = 0 $.

### Explanation of Why Other Options are Incorrect:

- **Option 2:** "If $ S_1 $ is not true, then $ S_2 $ is also not true." This is incorrect because $ S_2 $ is always true regardless of $ S_1 $. $ S_2 $ is a homogeneous equation and always represents lines through the origin, independent of the truth of $ S_1 $.

- **Option 3:** "$S_2$ is always true and $S_1$ implies $S_2$, if $c=0$." While it is true that $ S_2 $ is always true, the implication that $ S_1 $ implies $ S_2 $ only when $ c = 0 $ is misleading. $ S_2 $ is true regardless of $ S_1 $, and the condition $ c = 0 $ is not necessary for $ S_2 $ to be true.

- **Option 4:** "Both $ S_1 $ and $ S_2 $ imply each other." This is incorrect because $ S_2 $ is always true independently of $ S_1 $. $ S_1 $ does not necessarily imply $ S_2 $ unless specific conditions (like $ c = 0 $) are met, and even then, $ S_2 $ is true regardless.

In conclusion, Option 1 is correct because it accurately reflects the relationship between $ S_1 $ and $ S_2 $ under the given conditions. $ S_2 $ is always true, and if $ S_1 $ is true, $ S_2 $ is also true, particularly when $ c = 0 $.