Exercises: Parabola

Find the standard equation of the parabola with vertex at the origin of coordinates and focus at the point $F(0, 3)$.

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 $x = 4$.

Find the locus of points in the plane that are equidistant from the point $A(0, 4)$ and the horizontal line $y = -4$.

Calculate the coordinates of the points of intersection of the parabola $x^2 = 4y$ with the bisector line of the first quadrant $y = x$.

Identify whether the general equation $y^2 - 4y - 6x + 10 = 0$ corresponds to a parabola and, if so, find the coordinates of its vertex.

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?

Find the equation of the parabola with vertex at the origin, axis of symmetry coinciding with the x-axis, and passing through the point $P(-2, 4)$.

Álvaro has to calculate the focus and the directrix of the parabola whose standard equation is $y^2 = 12x$. On his exam he writes:

"Since the canonical form is $y^2 = 2px$, it follows immediately that the parameter is $p = 12$. Therefore, the focus is located at the point $F(12, 0)$ and the equation of the directrix is $x = -12$".

1. a) Identify the conceptual error that Álvaro made when identifying the parameter $p$ and its consequences on the elements of the parabola.
2. b) Explain in your own words what the parameter $p$ represents geometrically with respect to the distance between the focus and the directrix.
3. c) Correct the resolution, deriving in a justified manner the real value of $p$, the correct coordinates of the focus $F$, and the equation of the directrix.

Construct the canonical equation of a parabola with vertex at the origin $V(0, 0)$ knowing that its axis of symmetry coincides with the x-axis, that its branches open towards the negative values of $x$, and that the geometric distance between its focus and its directrix is exactly $8$ units of length.

Once the equation is obtained:

1. a) Determine the coordinates of its focus $F$ and the equation of its directrix $d$.
2. b) Find the coordinates of the point $P$ on the parabola whose ordinate is $y = 4$.
3. c) Prove numerically, by explicitly calculating the distances $d(P, F)$ and $d(P, d)$, that said point exactly satisfies the locus definition of the parabola.

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.

Find the equation of the tangent line to the parabola $y^2 = 8x$ at the point with y-coordinate $y = 4$.

A parabolic arch of a stone bridge has a span (width at the base) of $40\text{ m}$ and a maximum height of $10\text{ m}$. Calculate the height of the arch at a horizontal distance of $8\text{ m}$ from the centre of the base.

Find the locus of the centres of the circles that are tangent to the line $y = -2$ and pass simultaneously through the point $P(0, 2)$. Identify the resulting conic section and detail its main elements.

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.

Determine the equation of the parabola with a vertical axis passing through the points $A(-1, 0)$, $B(1, -2)$ and $C(3, 4)$.

Study the relative position of the line $r: 2x + y - 3 = 0$ and the parabola $y^2 = 4x$. If they intersect, find the exact length of the chord determined by the intersection of the line and the parabola.

Determine the value of the parameter $k$ so that the line $y = 2x + k$ is tangent to the parabola $x^2 = 2y$.

At the Plataforma Solar de Almería, parabolic trough collectors are used to concentrate direct solar radiation onto a receiver tube through which a thermal fluid flows. The cross-section of the reflector mirror has a parabolic shape symmetrical with respect to the vertical axis, with an upper aperture width of $5,6\text{ m}$ and a maximum depth at the center of $1,4\text{ m}$.

1. a) Place the vertex at the origin of coordinates $V(0, 0)$ and model the profile of the collector by determining its canonical equation of the form $x^2 = 2py$.
2. b) Apply the reflective property of conics to mathematically justify at what exact height above the bottom vertex the receiver tube must be fixed in order to absorb the maximum amount of solar radiation parallel to the axis.
3. c) If the directrix of the parabola represents a reference level for the lower mechanical structure, at what distance below the vertex is this line located?

Consider the parabola with translated vertex of equation $(y - 2)^2 = 4(x + 1)$. A focal chord perpendicular to its axis of symmetry is drawn (that is, the segment that passes through the focus, has its endpoints on the parabola, and is perpendicular to said axis).

1. a) Reasonably estimate the length of this focal chord before calculating the coordinates of its endpoints, relying exclusively on the definition of the parabola as a locus and on the distance from the focus to the directrix.
2. b) Analytically determine the coordinates of the vertex $V$, of the focus $F$, the equation of the directrix, and the coordinates of the endpoints $A$ and $B$ of said focal chord.
3. c) Calculate the exact Euclidean distance between $A$ and $B$, check whether it matches your previous estimate, and reason why for any parabola of parameter $p$ the length of said focal chord perpendicular to the axis (called "latus rectum") is always $2p$.

Find the equations of the tangent lines to the parabola $y^2 = 4x$ drawn from the external point $P(-1, 0)$.

Using the definition of a parabola as a locus, find the general equation of the parabola with focus at point $F(2, 3)$ and directrix given by the oblique line $x + y - 1 = 0$.

A water cannon launches a jet whose trajectory describes a parabola. The water leaves from the origin of coordinates and the jet reaches a maximum height of $15\text{ m}$ at a horizontal distance of $20\text{ m}$. Determine analytically whether the jet of water will manage to pass over a wall $10\text{ m}$ high located at a horizontal distance of $30\text{ m}$ from the launch point.

Given the parabola with equation $y^2 = 2px$, prove analytically that the locus of the orthogonal projections of the focus onto the infinitely many tangent lines to the parabola is precisely the tangent line at the vertex.

Find the equation of the circle passing through the vertex of the parabola $y^2 = 8x$ and through the two points of intersection of this parabola with the vertical line $x = 2$.

Determine the coordinates of the point on the parabola $x^2 = 4y$ that is at the minimum distance from the line $r: x - y - 4 = 0$. Also, calculate this minimum distance.

Consider a parabolic mirror whose cross-section corresponds to the equation $y^2 = 4x$. A ray of light, parallel to the x-axis, strikes the mirror at point $P(1, 2)$. Prove analytically that the line containing the reflected ray passes precisely through the focus of the parabola.

A parabola has its vertex at point $V(1, 2)$ and its axis of symmetry is parallel to the y-axis. Given that the parabola is tangent to the line $y = x - 1$, find the general equation of the parabola and the exact coordinates of its focus.

We want to determine the equation of the tangent line to the parabola $y^2 = 4x$ at the point $P(1, 2)$ belonging to it. To solve this problem, two conceptually different methods are proposed:

• Method A (Algebraic): Set up the family of lines passing through $P$, $y - 2 = m(x - 1)$, substitute into the equation of the parabola, and require that the resulting quadratic equation has a double root by imposing that its discriminant is zero ($\Delta = 0$).

• Method B (Geometric): Use the definition of the parabola. With $F$ being the focus and $D$ the orthogonal projection of $P$ onto the directrix, the tangent line is exactly the perpendicular bisector of the segment $FD$ (the line perpendicular to $FD$ passing through its midpoint).

1. a) Fully develop Method A and obtain the value of the slope $m$ and the tangent line.
2. b) Fully develop Method B: find the coordinates of $F$ and $D$, calculate the midpoint and the direction vector of the perpendicular bisector, and obtain the tangent line.
3. c) Critically compare both procedures: argue which one is algebraically more direct and which one provides a better understanding of the optical reflection property used by headlamps and radio telescopes.

At the Complejo Espacial de Maspalomas (Gran Canaria), a team of engineers is designing a parabolic satellite dish oriented towards the zenith (vertical axis). The circular upper aperture of the antenna has a diameter of $12\text{ m}$. Due to mechanical design specifications and the stability of the receiver mast, it is required that the focus of the parabola be located at exactly the same height as the upper perimeter rim of the dish.

1. a) Placing the vertex at the origin of coordinates $(0, 0)$, derive reasonably the value of the parameter $p$, the total depth of the antenna at its center, and write its standard equation $x^2 = 2py$.
2. b) An incident vertical signal ray strikes the reflective surface at a distance of $3\text{ m}$ from the axis of symmetry (at the point with abscissa $x = 3$). Find the slope of the tangent line at that point (using derivatives or by imposing a zero discriminant) and deduce the components of a normal vector perpendicular to the surface at the impact.
3. c) Prove analytically, using the dot product and the definition of the cosine of the angle between two vectors, that the angle of incidence formed by the vertical ray with the normal vector is identical to the angle of reflection formed by the direction toward the focus with said normal vector, proving the physical law of reflection for this case.