Parabola
Satellite dishes, car headlights and reflecting telescopes share a remarkable property: every ray that arrives parallel to the axis is reflected through exactly the same point, the focus. This property follows directly from how the parabola is defined as a locus.
The parabola as a locus
Find the standard equation of the parabola with vertex at the origin of coordinates and focus at the point $F(0, 3)$.
Find the standard equation of the parabola with vertex at the origin of coordinates and focus at the point $F(0, 3)$.
Elements and standard equation
Determine the coordinates of the vertex, the focus, and the equation of the directrix of the parabola with equation $y^2 = -10x$.
Determine the coordinates of the vertex, the focus, and the equation of the directrix of the parabola with equation $y^2 = -10x$.
Write the equation of the parabola whose vertex is the point $V(2, -1)$ and whose directrix is the line $x = 4$.
Write the equation of the parabola whose vertex is the point $V(2, -1)$ and whose directrix is the line $x = 4$.
Translated parabolas and the general equation
If the vertex is not at the origin, it is translated just as with the other conics: $(y-y_0)^2=2p(x-x_0)$. Expanding that expression always gives a first-degree term in the axis variable and a second-degree term in the other one: for example $y^2-4y-6x+10=0$.
Identify whether the general equation $y^2 - 4y - 6x + 10 = 0$ corresponds to a parabola and, if so, find the coordinates of its vertex.
Identify whether the general equation $y^2 - 4y - 6x + 10 = 0$ corresponds to a parabola and, if so, find the coordinates of its vertex.
Given the general equation of the parabola $x^2 - 6x - 8y + 25 = 0$, complete the square to express it in its translated standard form and determine the coordinates of…
Given the general equation of the parabola $x^2 - 6x - 8y + 25 = 0$, complete the square to express it in its translated standard form and determine the coordinates of its vertex, its focus, and the equation of its directrix.
The reflective property
The profile of a car halogen headlight has the shape of a parabola. If its equation relative to a frame of reference centred at the vertex is $y^2 = 12x$ (with…
The profile of a car halogen headlight has the shape of a parabola. If its equation relative to a frame of reference centred at the vertex is $y^2 = 12x$ (with measurements expressed in centimetres), at what distance from the vertex should the bulb be placed so that the light rays reflect parallel to the axis of symmetry?
The reflector of a parabolic antenna has an outer diameter of $2\text{ m}$ and a maximum depth of $0,25\text{ m}$. Determine at what distance from the vertex the focus…
The reflector of a parabolic antenna has an outer diameter of $2\text{ m}$ and a maximum depth of $0,25\text{ m}$. Determine at what distance from the vertex the focus is located where the signal receiver should be placed.