Answer :
Let's analyze each option step-by-step to find the equation of the parabola with its vertex at [tex]\((0,0)\)[/tex] and its focus along the negative part of the [tex]\(x\)[/tex]-axis.
### General Form of Parabola
1. For a Parabola opening to the left:
- The general equation is [tex]\( y^2 = -4ax \)[/tex], where [tex]\( a \)[/tex] is the distance from the vertex to the focus.
- Notice that this equation includes a negative coefficient for [tex]\( x \)[/tex].
### Analyzing Each Given Option
1. Option 1: [tex]\( y^2 = x \)[/tex]
- The equation [tex]\( y^2 = x \)[/tex] represents a parabola that opens to the right, with its vertex at [tex]\((0,0)\)[/tex].
- Since the parabola opens to the right and the focus is on the positive [tex]\(x\)[/tex]-axis, this does not match our condition of having the focus along the negative [tex]\(x\)[/tex]-axis.
2. Option 2: [tex]\( y^2 = -2x \)[/tex]
- The equation [tex]\( y^2 = -2x \)[/tex] fits the form [tex]\( y^2 = -4ax \)[/tex], where [tex]\( 4a = 2 \)[/tex] making [tex]\( a = \frac{1}{2} \)[/tex].
- This means the parabola opens to the left, placing the focus on the negative [tex]\( x \)[/tex]-axis.
- This matches our condition.
3. Option 3: [tex]\( x^2 = 4y \)[/tex]
- The equation [tex]\( x^2 = 4y \)[/tex] represents a parabola that opens upwards, with its vertex at [tex]\((0,0)\)[/tex].
- As the parabola opens upwards, the focus is on the positive [tex]\( y \)[/tex]-axis, which does not match our requirement.
4. Option 4: [tex]\( x^2 = -6y \)[/tex]
- The equation [tex]\( x^2 = -6y \)[/tex] fits the form [tex]\( x^2 = -4ay \)[/tex], where [tex]\( 4a = 6 \)[/tex] making [tex]\( a = \frac{3}{2} \)[/tex].
- This means the parabola opens downwards, placing the focus on the negative [tex]\( y \)[/tex]-axis.
- Since this does not match our requirement as the focus should be on the negative [tex]\( x \)[/tex]-axis, this is not correct.
### Conclusion
The correct option is:
[tex]\[ y^2 = -2x \][/tex]
This equation represents a parabola with its vertex at [tex]\((0,0)\)[/tex] and its focus along the negative part of the [tex]\(x\)[/tex]-axis.
### General Form of Parabola
1. For a Parabola opening to the left:
- The general equation is [tex]\( y^2 = -4ax \)[/tex], where [tex]\( a \)[/tex] is the distance from the vertex to the focus.
- Notice that this equation includes a negative coefficient for [tex]\( x \)[/tex].
### Analyzing Each Given Option
1. Option 1: [tex]\( y^2 = x \)[/tex]
- The equation [tex]\( y^2 = x \)[/tex] represents a parabola that opens to the right, with its vertex at [tex]\((0,0)\)[/tex].
- Since the parabola opens to the right and the focus is on the positive [tex]\(x\)[/tex]-axis, this does not match our condition of having the focus along the negative [tex]\(x\)[/tex]-axis.
2. Option 2: [tex]\( y^2 = -2x \)[/tex]
- The equation [tex]\( y^2 = -2x \)[/tex] fits the form [tex]\( y^2 = -4ax \)[/tex], where [tex]\( 4a = 2 \)[/tex] making [tex]\( a = \frac{1}{2} \)[/tex].
- This means the parabola opens to the left, placing the focus on the negative [tex]\( x \)[/tex]-axis.
- This matches our condition.
3. Option 3: [tex]\( x^2 = 4y \)[/tex]
- The equation [tex]\( x^2 = 4y \)[/tex] represents a parabola that opens upwards, with its vertex at [tex]\((0,0)\)[/tex].
- As the parabola opens upwards, the focus is on the positive [tex]\( y \)[/tex]-axis, which does not match our requirement.
4. Option 4: [tex]\( x^2 = -6y \)[/tex]
- The equation [tex]\( x^2 = -6y \)[/tex] fits the form [tex]\( x^2 = -4ay \)[/tex], where [tex]\( 4a = 6 \)[/tex] making [tex]\( a = \frac{3}{2} \)[/tex].
- This means the parabola opens downwards, placing the focus on the negative [tex]\( y \)[/tex]-axis.
- Since this does not match our requirement as the focus should be on the negative [tex]\( x \)[/tex]-axis, this is not correct.
### Conclusion
The correct option is:
[tex]\[ y^2 = -2x \][/tex]
This equation represents a parabola with its vertex at [tex]\((0,0)\)[/tex] and its focus along the negative part of the [tex]\(x\)[/tex]-axis.
