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Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.?
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Equation of a Right Circular Cone
To find the equation of a right circular cone, we need to consider the properties of a cone and the given information about its vertex and base.
Properties of a Right Circular Cone:
- A cone is a three-dimensional geometric figure with a circular base and a pointed top called the vertex.
- The axis of a cone is a line segment joining the vertex and the center of the base.
- The height of a cone is the perpendicular distance from the vertex to the base.
- The slant height of a cone is the distance from the vertex to any point on the circumference of the base.
Given Information:
- The vertex of the cone is at the origin, which means the coordinates of the vertex are (0, 0, 0).
- The base of the cone is a circle with the equation x = a, where 'a' is a constant.
- The equation of the base circle in terms of y and z is given by y^2 z^2 = b^2, where 'b' is a constant.
Equation of the Cone:
Let's derive the equation of the right circular cone based on the given information.
1. Equation of the Base Circle:
The equation of the base circle is x = a. Since the vertex is at the origin, the coordinates of any point on the base circle can be represented as (a, y, z).
2. Equation of the Slant Height:
The slant height of the cone is the distance from the vertex to any point on the circumference of the base. Using the distance formula in three dimensions, the equation of the slant height can be written as:
√(x^2 + y^2 + z^2) = h, where h is the height of the cone.
3. Combining the Equations:
To find the equation of the right circular cone, we need to combine the equation of the base circle and the equation of the slant height. Substituting the values of x, y, and z from the equation of the base circle into the equation of the slant height, we get:
√(a^2 + y^2 + z^2) = h
4. Final Equation:
Squaring both sides of the equation, we get:
a^2 + y^2 + z^2 = h^2
Therefore, the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y^2 z^2 = b^2 is:
a^2 + y^2 + z^2 = h^2
To find the equation of a right circular cone, we need to consider the properties of a cone and the given information about its vertex and base.
Properties of a Right Circular Cone:
- A cone is a three-dimensional geometric figure with a circular base and a pointed top called the vertex.
- The axis of a cone is a line segment joining the vertex and the center of the base.
- The height of a cone is the perpendicular distance from the vertex to the base.
- The slant height of a cone is the distance from the vertex to any point on the circumference of the base.
Given Information:
- The vertex of the cone is at the origin, which means the coordinates of the vertex are (0, 0, 0).
- The base of the cone is a circle with the equation x = a, where 'a' is a constant.
- The equation of the base circle in terms of y and z is given by y^2 z^2 = b^2, where 'b' is a constant.
Equation of the Cone:
Let's derive the equation of the right circular cone based on the given information.
1. Equation of the Base Circle:
The equation of the base circle is x = a. Since the vertex is at the origin, the coordinates of any point on the base circle can be represented as (a, y, z).
2. Equation of the Slant Height:
The slant height of the cone is the distance from the vertex to any point on the circumference of the base. Using the distance formula in three dimensions, the equation of the slant height can be written as:
√(x^2 + y^2 + z^2) = h, where h is the height of the cone.
3. Combining the Equations:
To find the equation of the right circular cone, we need to combine the equation of the base circle and the equation of the slant height. Substituting the values of x, y, and z from the equation of the base circle into the equation of the slant height, we get:
√(a^2 + y^2 + z^2) = h
4. Final Equation:
Squaring both sides of the equation, we get:
a^2 + y^2 + z^2 = h^2
Therefore, the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y^2 z^2 = b^2 is:
a^2 + y^2 + z^2 = h^2
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Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.?
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Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.?.
Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equation of the right circular cone whose vertex is at the origin and base is the circle x = a, y ^2 z^2= b^2.?.
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