find the equation of thecone whose vertex is (α, β , γ) and guiding curve is y^2=4ax, z=0
Answers
Question:
Find the equation of thecone whose vertex is (α, β , γ) and guiding curve is y^2=4ax, z=0
Answer:
Here's your answer.
Step-by-step explanation:
If the cone base is an ellipse and its vertex is at origin, then the equation is:
x²/a² + y²/b² = z² / c²
So the base elliptical surface is on x-y plane and z axis is the axis of cone.
If the base is a circle, then a = b and hence, for a right circular cone:
x² + y² = (a²/c²) z² = k z²
The general equation of a cone in three dimensions with an elliptical base and an axis inclined to x, y and z axes :
a x² + b y² + c z² + d x + e y + f z + g x y + h y z + i z x + k = 0
An equation of 2nd degree in x, y and z, if it passes through (0,0,0) then:
a x² + b y² + c z² + d x + e y + f z + g x y + h y z + i z x = 0
Hope you understand.
Answer:
The equation of the cone with vertex (α, β, γ) and guiding curve y^2 = 4ax, z = 0 is:
(x - α)^2 + (4a - y)^2 + γ^2 = k^2
Step-by-step explanation:
To find the equation of the cone with vertex (α, β, γ) and guiding curve y^2 = 4ax, z = 0, we can start by considering the distance between the vertex and a point on the curve. Let (x, y, 0) be a point on the curve. Then, the distance between this point and the vertex is given by:
sqrt((x - α)^2 + (y - β)^2 + γ^2)
Since the point lies on the curve, we can substitute y^2 = 4ax into the above equation and simplify to obtain:
sqrt((x - α)^2 + (4a - y)^2 + γ^2)
This expression represents the distance between the vertex and any point on the cone. To obtain the equation of the cone, we need to set this expression equal to a constant value k and solve for x, y, and z. This gives us the general equation of a cone:
(x - α)^2 + (4a - y)^2 + γ^2 = k^2
Therefore, the equation of the cone with vertex (α, β, γ) and guiding curve y^2 = 4ax, z = 0 is:
(x - α)^2 + (4a - y)^2 + γ^2 = k^2
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