Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine which of the given equations corresponds to a conic section formed when a plane intersects a cone parallel to the base, let’s analyze each given equation:
1. [tex]\( x^2 + y^2 = 3^2 \)[/tex]
2. [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]
3. [tex]\( x^2 = 8 y \)[/tex]
4. [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]
### Step-by-Step Analysis:
1. Equation [tex]\( x^2 + y^2 = 3^2 \)[/tex]:
- This equation is in the form of [tex]\( x^2 + y^2 = r^2 \)[/tex], which represents a circle centered at the origin with radius [tex]\(\sqrt{9} = 3\)[/tex].
- Conic section: Circle
2. Equation [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]:
- This equation is in the form of [tex]\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)[/tex], which represents an ellipse centered at the origin with semi-major axis 3 and semi-minor axis 2.
- Conic section: Ellipse
3. Equation [tex]\( x^2 = 8y \)[/tex]:
- This equation is in the form of [tex]\( x^2 = 4ay \)[/tex], which represents a parabola that opens upwards.
- Conic section: Parabola
4. Equation [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]:
- This equation is in the form of [tex]\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)[/tex], which represents a hyperbola centered at the origin.
- Conic section: Hyperbola
### Conclusion:
- A circle is specifically formed when a plane intersects a cone parallel to the base of the cone.
- Among the given equations, [tex]\( x^2 + y^2 = 3^2 \)[/tex] represents a circle.
Therefore, the correct equation corresponding to a conic section formed when a plane intersects a cone parallel to the base is:
[tex]\[ x^2 + y^2 = 3^2 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{1} \][/tex]
1. [tex]\( x^2 + y^2 = 3^2 \)[/tex]
2. [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]
3. [tex]\( x^2 = 8 y \)[/tex]
4. [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]
### Step-by-Step Analysis:
1. Equation [tex]\( x^2 + y^2 = 3^2 \)[/tex]:
- This equation is in the form of [tex]\( x^2 + y^2 = r^2 \)[/tex], which represents a circle centered at the origin with radius [tex]\(\sqrt{9} = 3\)[/tex].
- Conic section: Circle
2. Equation [tex]\( \frac{x^2}{2^2} + \frac{y^2}{3^2} = 1 \)[/tex]:
- This equation is in the form of [tex]\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)[/tex], which represents an ellipse centered at the origin with semi-major axis 3 and semi-minor axis 2.
- Conic section: Ellipse
3. Equation [tex]\( x^2 = 8y \)[/tex]:
- This equation is in the form of [tex]\( x^2 = 4ay \)[/tex], which represents a parabola that opens upwards.
- Conic section: Parabola
4. Equation [tex]\( \frac{x^2}{2^2} - \frac{y^2}{3^2} = 1 \)[/tex]:
- This equation is in the form of [tex]\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)[/tex], which represents a hyperbola centered at the origin.
- Conic section: Hyperbola
### Conclusion:
- A circle is specifically formed when a plane intersects a cone parallel to the base of the cone.
- Among the given equations, [tex]\( x^2 + y^2 = 3^2 \)[/tex] represents a circle.
Therefore, the correct equation corresponding to a conic section formed when a plane intersects a cone parallel to the base is:
[tex]\[ x^2 + y^2 = 3^2 \][/tex]
So, the correct choice is:
[tex]\[ \boxed{1} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.