Feodalherren said:Homework Statement
Asymptotes at y=±[itex]\sqrt{10}[/itex] x /5
Foci: (±[itex]\sqrt{7}[/itex],0)
Homework Equations
The Attempt at a Solution
It's obviously a hyperbola. What I can't wrap my head around is why a^2 + b^2 ≠ c^2
No. c = √7.Feodalherren said:c = sqrt 7 ; correct?
Feodalherren said:This yields
7=25+10
obviously NOT true. What am I missing here?
Feodalherren said:Sorry I meant c^2 = 7.
Still. The asymptotes of a hyperbola that shoots of in the x direction is positive and negative a/b.
I don't get this at all.
A conic section is a curve formed by the intersection of a plane and a cone. It is important because it has many practical applications in mathematics, physics, and engineering, such as in the study of orbits, reflection and refraction of light, and the design of structures and machines.
The four main types of conic sections are the circle, ellipse, parabola, and hyperbola. These are distinguished by their shapes and mathematical equations.
The equation of a conic section depends on its type and the specific characteristics of the curve. For example, the equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. The equation of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the semi-major and semi-minor axes. The equations for a parabola and hyperbola are more complex and involve parameters such as the distance from the focus and the slope of the directrix.
Conic sections can be found in many natural and man-made objects and phenomena. Some examples include the orbits of planets and satellites, the shape of car headlights and satellite dishes, the path of a thrown ball, and the cross-section of a tree trunk or a seashell.
Knowing the equation of a conic section can help us understand and analyze its properties and behavior. This information can be used in various fields, such as astronomy, engineering, and architecture, to design and optimize structures and systems. It also allows us to make predictions and solve problems involving conic sections.