Parabola
A satellite dish on a rooftop in Santiago, the headlight of a pickup truck, and the arch of a bridge have something in common: their shape is a parabola. In dishes and headlights, every ray parallel to the axis passes through a single point, the focus. Why does this happen? The answer comes from how the parabola is defined as a locus.
The parabola as a locus
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.
Elements and standard equation
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 $P(-2, 4)$.
Translated parabolas and the general equation
If the vertex is $V(h,k)$, the parabola is translated just like the other conics:
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Horizontal axis: $(y-k)^2=2p(x-h)$, focus $\left(h+\frac{p}{2},k\right)$, directrix $x=h-\frac{p}{2}$.
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Vertical axis: $(x-h)^2=2p(y-k)$, focus $\left(h,k+\frac{p}{2}\right)$, directrix $y=k-\frac{p}{2}$.
Expanding gives the general equation, with one squared variable and the other of degree 1, for example $y^2-4y-6x+10=0$.
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.
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.
Parabolas through points and real-world problems
A parabola with a vertical axis can be written as $y=ax^2+bx+c$. With three points you set up a system of three equations. If you know the vertex, it is shorter to use $y-k=a(x-h)^2$ and substitute a single point.
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.
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 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.
Distance from a point to a line, orthogonal projection, and slanted directrices
When the directrix is neither horizontal nor vertical, the distance from $P(x,y)$ to it is not as simple as $|y+2|$. You need the formula for the distance from a point to a line.
With it, the definition $d(P,F)=d(P,d)$ works with any directrix. The resulting equation has an $xy$ term (the axis of the parabola is tilted).
The same formula can be used to find the point on a parabola closest to a line.
Orthogonal projection of a point onto a line
To find $H$:
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Find the slope of $r$ and that of the perpendicular (from $m$ to $-\frac1m$; if $r$ is $Ax+By+C=0$, the perpendicular has direction vector $(A,B)$).
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Write the perpendicular line through $P$.
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Solve the system formed by $r$ and that perpendicular: the solution is $H$.
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 $x + y - 1 = 0$.
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.
Position of lines and tangents
To find out how a line meets a parabola, substitute the line into the equation of the parabola and analyze the resulting quadratic equation:
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discriminant $>0$: secant (two points);
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discriminant $=0$: tangent (one point);
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discriminant $<0$: exterior (none).
When the line is a secant, the chord is the segment between the two intersection points; its length is found with the distance formula between two points.
To draw tangents from an exterior point, set up the line $y=mx+n$ through the point and require the discriminant to be zero.
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 $P(-1, 0)$.
A circle through points on the parabola
Some problems combine the parabola with a circle. Recall the general equation of the circle (you study it in the Circle section):
$$x^2+y^2+Dx+Ey+F=0$$with center $\left(-\dfrac D2,-\dfrac E2\right)$. It has three unknowns ($D$, $E$, $F$): each point it passes through gives one equation.
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.
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.
Back to the beginning: the dish and the headlight work because the tangent at each point makes equal angles with the parallel ray and with the segment to the focus; that way all the rays meet at $F$.