The Parabola
Satellite dishes that pick up TV signals, car headlights, and reflecting telescopes share an amazing property: every ray that arrives parallel to the axis is reflected through exactly the same point, the focus. Why does this happen, and where do you have to put the receiver? It all comes from how the parabola is defined as a locus.
The parabola as a locus
Distance from a point to a line
When the directrix is horizontal or vertical, the distance is just a subtraction. If the directrix is slanted, you need a formula.
Find the locus of points in the plane that are equidistant from the point $A(0, 4)$ and the horizontal line $y = -4$.
Find the locus of points in the plane that are equidistant from the point $A(0, 4)$ and the horizontal line $y = -4$.
Find the locus of the centers of the circles that are tangent to the line $y = -2$ and pass simultaneously through the point $P(0, 2)$. Identify the resulting conic…
Find the locus of the centers 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.
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…
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 $x + y - 1 = 0$.
Elements and standard equation with vertex at the origin
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 $F(0, 3)$.
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 $y^2 = -10x$.
Find the equation of the parabola with vertex at the origin, axis of symmetry coinciding with the x-axis, and passing through the point $P(-2, 4)$.
Find the equation of the parabola with vertex at the origin, axis of symmetry coinciding with the x-axis, and passing through the point $P(-2, 4)$.
Parabolas with vertex away from the origin
If the vertex is $V(h,k)$, the curve is translated: the standard equation replaces $x$ with $x-h$ and $y$ with $y-k$.
When expanded, a first-degree term appears in the variable along the axis and a second-degree term in the other one. For example, $y^2-4y-6x+10=0$ is a general equation of a parabola: only one of the variables is squared, and it is converted to standard form by completing the square.
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 $x = 4$.
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.
Determine the equation of the parabola with a vertical axis passing through the points $A(-1, 0)$, $B(1, -2)$ and $C(3, 4)$.
Determine the equation of the parabola with a vertical axis passing through the points $A(-1, 0)$, $B(1, -2)$ and $C(3, 4)$.
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…
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.
Lines and parabolas: intersection, tangents, and minimum distance
To find out how a line is positioned relative to a parabola, substitute the line into the parabola and you get a quadratic equation. Its discriminant $D=b^2-4ac$ decides:
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$D>0$: the line is a secant (two points).
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$D=0$: it is tangent (one point).
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$D<0$: it is outside (they do not meet at real points).
You can also find the tangent at a point using the derivative, or with the tangent formula at $(x_1,y_1)$ for $y^2=4px$: $y\,y_1=2p(x+x_1)$.
Calculate the coordinates of the points of intersection of the parabola $x^2 = 4y$ with the bisector line of the first quadrant $y = x$.
Calculate the coordinates of the points of intersection of the parabola $x^2 = 4y$ with the bisector line of the first quadrant $y = x$.
Find the equation of the tangent line to the parabola $y^2 = 8x$ at the point with y-coordinate $y = 4$.
Find the equation of the tangent line to the parabola $y^2 = 8x$ at the point with y-coordinate $y = 4$.
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…
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 $x^2 = 2y$.
Determine the value of the parameter $k$ so that the line $y = 2x + k$ is tangent to the parabola $x^2 = 2y$.
Find the equations of the tangent lines to the parabola $y^2 = 4x$ drawn from the external point $P(-1, 0)$.
Find the equations of the tangent lines to the parabola $y^2 = 4x$ drawn from the external point $P(-1, 0)$.
Point on a parabola closest to a line
A point on the parabola is written with a parameter, the distance to the line is expressed with the point-to-line formula, and you get a quadratic in the parameter: its minimum is at the vertex (by completing the square).
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.
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.
Circle through three points
A circle with center $(h,k)$ and radius $r$ has equation $(x-h)^2+(y-k)^2=r^2$. Expanding it gives the general form $x^2+y^2+Dx+Ey+F=0$, with three unknowns $D$, $E$, $F$: three points on the circle give three equations. Then the center is $\left(-\tfrac D2,-\tfrac E2\right)$ and $r^2=h^2+k^2-F$.
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…
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 $x = 2$.
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 centered 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 centered at the vertex is $y^2 = 12x$ (with measurements expressed in centimeters), 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.
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 $P(1, 2)$.…
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.
Modeling with parabolas
Bridge arches, the path of a ball, or a stream of water are parabolas. The method: place the coordinate system somewhere convenient (almost always with the vertex at the origin or at the highest point), write the equation with a parameter, and use a known point to find it.
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…
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 center of the base.
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…
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.
Projection of a point onto a line
The orthogonal projection of a point $F$ onto a line $r$ is the foot of the perpendicular: the point where the perpendicular to $r$ drawn through $F$ meets $r$. To find it: (1) find the slope of the perpendicular, which is the negative reciprocal of the slope of $r$; (2) write the perpendicular line through $F$; (3) solve the system formed by the two lines.
This is useful for proving properties of loci: you describe the family of tangents with a parameter, project the focus onto each one, and see what relationship the foot satisfies.
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…
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.