The curve `y^(2) = -4ax` where, `(a gt 0) ` lies in.

Options

A. First and fourth quadrants
B. First and second quadrants
C. Second and third quadrants
D. Third and fourth quadrants

Answer

To determine in which quadrant the curve defined by the equation \( y^2 = -4ax \) (where \( a > 0 \)) lies, we can follow these steps:

### Step 1: Analyze the Equation
The given equation is \( y^2 = -4ax \). Here, \( a \) is a positive constant. This indicates that the right side of the equation is negative for all values of \( x \) since \( -4a \) is negative when \( a > 0 \).

### Step 2: Determine the Sign of \( y^2 \)
Since \( y^2 \) is always non-negative (i.e., \( y^2 \geq 0 \)), the equation \( y^2 = -4ax \) implies that the left side must also be non-negative. Therefore, for the equation to hold true, the right side must also be non-negative.

### Step 3: Analyze the Implications
For \( -4ax \) to be non-negative, \( x \) must be non-positive (i.e., \( x \leq 0 \)). This means that the curve can only exist in the left half of the Cartesian plane.

### Step 4: Determine the Values of \( y \)
Since \( y^2 \) is non-negative, \( y \) can take both positive and negative values. Therefore, for each non-positive \( x \), there are corresponding positive and negative values of \( y \).

### Step 5: Identify the Quadrants
Given that \( x \leq 0 \) and \( y \) can be both positive and negative, the curve will lie in:
- The second quadrant (where \( x < 0 \) and \( y > 0 \))
- The third quadrant (where \( x < 0 \) and \( y < 0 \))

### Conclusion
Thus, the curve defined by the equation \( y^2 = -4ax \) lies in the second and third quadrants.