Lesson A circle of maximum radius tangent to and inscribed in parabola

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<H2>OVERVIEW of lessons on parabolas</H2>

This file is your guide on my lessons on &nbsp;Parabolas&nbsp; under the topic &nbsp;Conic sections&nbsp; in this site. &nbsp;These lessons are:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-definition--canonical-equation--characteristic-points-and-elements.lesson>Parabola definition, canonical equation, characteristic points and elements</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-focal-property.lesson>Parabola focal property</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-to-a-parabola.lesson>Tangent lines and normal vectors to a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-a-parabola.lesson>Optical property of a parabola</A>.

In the first lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-definition--canonical-equation--characteristic-points-and-elements.lesson>Parabola definition, canonical equation, characteristic points and elements</A>&nbsp; the basic notions are introduced related to parabolas, 
such as &nbsp;canonical equation of a parabola, &nbsp;a focus of a parabola, a focus distance of a parabola, &nbsp;and a parabola directrix.

<TABLE>
 <TR>
 <TD>
In this lesson, &nbsp;parabola is defined as a curve in a plane 
such that all its points &nbsp;{({{{x}}},{{{y}}})}&nbsp; satisfy the equation&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

{{{y}}} = {{{x^2/(2p)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

with real positive number &nbsp;{{{p}}} > {{{0}}} &nbsp;in some rectangular 
coordinate system &nbsp;OXY&nbsp; in the plane (see &nbsp;Figure 1). 
 </TD>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing(242, 200, -3.0, 3.1, -2.5, 2.5,
 graph(242, 200, -3.0, 3.1, -2.5, 2.5,
 x^2/4,
 (x^2+(y-1)^2-0.005)<0,
 -1),
 locate (0.2, 1.3, F)
)}}}

Figure 1. &nbsp;The parabola, its focus and its directix
 </TD>
 </TR>
</TABLE>

In the second lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-focal-property.lesson>Parabola focal property</A>&nbsp; the focal property of a parabola is introduced and proved. 
The focal property of a parabola states that

<BLOCKQUOTE>A curve on a plane is a parabola if and only if the distance from any point of the curve to the fixed point on the plane (focus) 
is equal to the distance to the fixed straight line on the plane which does not pass through the focus.</BLOCKQUOTE> <TABLE> 
 <TR>
 <TD>
The focal property of a parabola is the characteristic property.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
Based on this property, one can define a parabola as a curve 
in a plane such that for any point of the curve the distance 
to the fixed point on the plane is equal to the distance to 
the fixed straight line on the plane not passing through the 
given fixed point.

This definition is equivalent to the algebraic definition 
of parabolas of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-definition--canonical-equation--characteristic-points-and-elements.lesson>Parabola definition, canonical equation, 
characteristic points and elements</A>.

So, the two definitions describe actually the same class of curves 
in a plane.

</TD>
 <TD>
{{{drawing(242, 200, -3.0, 3.1, -2.5, 2.5,
 graph(242, 200, -3.0, 3.1, -2.5, 2.5,
 x^2/4,
 (x^2+(y-1)^2-0.005)<0,
 -1),
 locate (0.2, 1.3, F),

line( 0,    1, -2.5, 1.5625),
          locate (-1.5, 1.8, r1),

line(-2.5, -1, -2.5, 1.5625),
          locate (-2.45, 0.8, r2),

locate (-2.5,   1.97, M),
          locate (-2.55, -1.05, X)
)}}}

Figure 2. &nbsp;To the parabola focal property

</TD>
 </TR>
</TABLE>

In the third lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-to-a-parabola.lesson>Tangent lines and normal vectors to a parabola</A>&nbsp; the formula is derived for a tangent line to a parabola in a plane.

<BLOCKQUOTE>Tangent line to a parabola &nbsp;&nbsp;&nbsp;{{{y = (1/2p)x^2}}}&nbsp; at the point &nbsp;({{{x[0]}}},{{{y[0]}}})&nbsp; has the equation &nbsp;{{{y-y[0] = (1/p)(x-x[0])*x[0]}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)</BLOCKQUOTE>

Usually, deriving this kind of formulas requires &nbsp;Calculus. &nbsp;In the lesson I am talking about I didn't use &nbsp;Calculus. 
I simply checked that the straight line &nbsp;(2)&nbsp; passes through the given point &nbsp;({{{x[0]}}},{{{y[0]}}})&nbsp; and have only one common point with the circle. 
This is enough for a straight line to be a tangent line to the smooth convex figure as a parabola is.

A corollary is derived from the formula &nbsp;(2). &nbsp;It relates to the normal vector of a tangent line to a parabola.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The "inward" normal vector to the tangent line of a parabola &nbsp;{{{y = (1/2p)x^2}}}&nbsp; at the point &nbsp;({{{x[0]}}},{{{y[0]}}})&nbsp; is &nbsp;({{{-x[0]/p}}},{{{1}}}).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The unit "inward" normal vector to the tangent line of a parabola &nbsp;{{{y = (1/2p)x^2}}}&nbsp; at the point &nbsp;({{{x[0]}}},{{{y[0]}}})&nbsp; is &nbsp;({{{-x[0]/N}}},{{{p/N}}}), &nbsp;where {{{N}}} = {{{sqrt(x[0]^2 + p^2)}}}.
.

In the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-a-parabola.lesson>Optical property of a parabola</A>&nbsp; the optical property of a parabola is considered and proved.

<TABLE><TR><TD>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Optical property of a parabola reads as follows &nbsp;(Figure 3):

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If to put the source of light into the focus of a parabola&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and if the internal surface of the parabola reflects the light rays 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;as a mirror, then after reflection all the light rays will be directed parallel 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;to the parabola axis.

Figure 3 &nbsp;displays the parabola with the focus point &nbsp;F. &nbsp;A source of light is placed 
at the focus &nbsp;F. &nbsp;Light rays emitted by the source after reflecting all move forward 
parallel to the parabola axis.

</TD>
 <TD>{{{drawing( 242, 200, -6.0, 6.1, -3.0, 7.0,
 graph ( 242, 200, -6.0, 6.1, -3.0, 7.0,
 x^2/4),

red(line ( 0, 1,  2, 1)),
        red(line ( 0, 1, -2, 1)),

line ( 2, 1,  2, 6.5),
            line (-2, 1, -2, 6.5),

red(line ( 0, 1,  2.828, 2)),
        red(line ( 0, 1, -2.828, 2)),

line ( 2.828, 2,  2.828, 6.5),
            line (-2.828, 2, -2.828, 6.5),

red(line (0,  1,  4, 4)),
        red(line (0,  1, -4, 4)),

line ( 4, 4,  4, 6.5),
            line (-4, 4, -4, 6.5),

circle (0, 1, 0.16)
)}}}

Figure 3. &nbsp;Optical property of a parabola
 </TD>
 </TR>
</TABLE>
This optical property is equivalent to any of the following geometric facts &nbsp;(Figure 4):

<TABLE> 
 <TR>
 <TD>
1. &nbsp;For any parabola's point the angle between the tangent line to the parabola&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;at this point and the focal radius to the point is congruent to the angle between 
&nbsp;&nbsp;&nbsp;the tangent line and the parabola axis.

2. &nbsp;For any parabola's point the angle between the normal to the parabola at this 
&nbsp;&nbsp;&nbsp;point and the focal radius to the point is congruent to the acute angle
&nbsp;&nbsp;&nbsp;between the normal and the parabola axis.

3. &nbsp;For any parabola's point the normal to the parabola at this point bisects 
&nbsp;&nbsp;&nbsp;the angle between the focal radius to the point and the straight ray drawn 
&nbsp;&nbsp;&nbsp;from the point parallel to the parabola axis.

</TD>
 <TD>{{{drawing( 242, 200, -6.0, 6.1, -3.0, 7.0,
 graph ( 242, 200, -6.0, 6.1, -3.0, 7.0,
 x^2/4,
 2x-4),

red(line (0,  1,  4, 4)),

line (4,  4,  4, 6.8),

red(line (4,  4,  8, 7)),

circle (0, 1, 0.16),

blue(line (4.0,   4.0,   2.211, 4.894)),
 blue(line (2.211, 4.896, 2.77,  4.83)),
 blue(line (2.211, 4.894, 2.60,  4.50)),

locate (-1.0, 1.35, F),
            locate ( 4.1, 4.2,  M),

arc (4.0, 4.0, 2.0, 2.0, 122, 147),
            arc (4.0, 4.0, 2.4, 2.4, 122, 147),
            locate (2.7, 3.2, alpha),

arc (4.0, 4.0, 2.0, 2.0, 272, 295),
            arc (4.0, 4.0, 2.4, 2.4, 272, 295),
            locate (4.07, 6.0, beta),

arc (4.0, 4.0, 1.6, 1.6, 150, 208),
            locate (2.7, 4.4, gamma),

arc (4.0, 4.0, 1.6, 1.6, 212, 268),
            locate (3.2, 5.4, delta),

line ( 4,   -1, 4,    3.8),

arc (4.0, 4.0, 2.4, 2.4, 300, 324),
            arc (4.0, 4.0, 2.8, 2.8, 300, 324),
            locate (5.07, 6.0, phi),

arc (4.0, 4.0, 2.4, 2.4, 90, 120),
            arc (4.0, 4.0, 2.8, 2.8, 90, 120),
            locate (3.3, 2.5, theta)
)}}}

Figure 4. &nbsp;To the optical property of 
a parabola &nbsp;&nbsp;({{{alpha}}}={{{beta}}}, {{{gamma}}}={{{delta}}}, {{{beta}}}={{{phi}}}, {{{alpha}}}={{{theta}}})
 </TD>
 </TR>
</TABLE>4. &nbsp;For any parabola's point the tangent line to the parabola at this point bisects the angle between the continuation of the focal vector 
&nbsp;&nbsp;&nbsp;to the point and the straight line passing through the point parallel to the parabola axis.

Figure 4 &nbsp;displays the parabola with the focus point &nbsp;F=({{{0}}},{{{p/2}}}), &nbsp;where &nbsp;{{{p}}}&nbsp; is the &nbsp;focal distance &nbsp;of the parabola. &nbsp;The focus is connected by the focal vector with

the point &nbsp;M&nbsp; at the parabola, which is chosen by an arbitrary way. &nbsp;The tangent line and the inward normal vector at the point &nbsp;M&nbsp; are displayed among with the straight

line drawn parallel to the parabola axis. &nbsp;The focal radius is continued. &nbsp;The optical property says that

&nbsp;&nbsp;&nbsp;&nbsp;- the angle &nbsp;{{{alpha}}}&nbsp; between the focal vector and the tangent line is congruent to the angle &nbsp;{{{beta}}}&nbsp; between the tangent line and the straight line passing 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;through the point parallel to the parabola axis: {{{alpha}}}={{{beta}}};

&nbsp;&nbsp;&nbsp;&nbsp;- the angle &nbsp;{{{gamma}}}&nbsp; between the focal vector and the normal line is congruent to the angle &nbsp;{{{delta}}}&nbsp; between the normal line and the straight line passing 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;through the point parallel to the parabola axis: &nbsp;{{{gamma}}}={{{delta}}}.

&nbsp;&nbsp;&nbsp;&nbsp;- the angle &nbsp;{{{beta}}}&nbsp; between the tangent line and the straight line passing through the point parallel to the parabola axis is congruent to the angle &nbsp;{{{phi}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;between the tangent line and the continuation of the focal vector: &nbsp;{{{beta}}}={{{phi}}}. &nbsp;Angles &nbsp;{{{alpha}}}&nbsp; and &nbsp;{{{theta}}}&nbsp; of the &nbsp;Figure 4&nbsp; are congruent too: {{{alpha}}}={{{theta}}}.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Actually, &nbsp;all four angles {{{alpha}}}, {{{beta}}}, {{{phi}}} and {{{theta}}} of the &nbsp;Figure 4&nbsp; are congruent: &nbsp;{{{alpha}}}={{{theta}}}={{{beta}}}={{{phi}}}.

Recall the physical &nbsp;law of reflection: &nbsp;the angle of incidence is equal to the angle of reflection measured form the normal. &nbsp;It is consistent with the geometric optical

properties above.

The proof of the optical properties is fully elementary. &nbsp;It uses the explicit formulas for the focal vector &nbsp;r1&nbsp; (Figure 4), &nbsp;including formulas for

its length from the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-focal-property.lesson>Parabola focal property</A>, &nbsp;and explicit formula for the parabola's normal vector of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-to-a-parabola.lesson>Tangent lines to a parabola</A>.

Note that 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If to reverse the direction of the light beams and to direct parallel rays of light onto a parabolic mirror along its axis,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;then all the light rays will be collected after reflection at the parabola focus.

This statement does not require a separate proof. &nbsp;It is fully supported by the geometric properties above and, therefore, is just proved.

Two bright spots are central in this series of lessons. &nbsp;The first is the elementary proof of equivalency the algebraic and the geometric definitions of a parabola

(the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-focal-property.lesson>Parabola focal property</A>). &nbsp;The second is the elementary proof of the optical property of a parabola &nbsp;(the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-a-parabola.lesson>Optical property of a parabola</A>).

By combining the &nbsp;parabola focal property&nbsp; and the &nbsp;parabola optical property&nbsp; we can formulate even stronger statement:

&nbsp;&nbsp;&nbsp;&nbsp;If to put the source of light into the parabola's focus point, and 
&nbsp;&nbsp;&nbsp;&nbsp;if the internal surface of the parabola reflects the light rays as a mirror, and 
&nbsp;&nbsp;&nbsp;&nbsp;if the speed of light is constant for the media filling the parabola interior, then
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- after reflection, all the light rays emitted by the source will be directed parallel to the parabola's axis, and 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- after reflection, the wave front for all the light rays emitted by the source at the same time moment will be a straight line 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;parallel to the parabola directrix.

In other words, after reflection the wave front of the reflected light rays is flat.

In order to this statement would be true, &nbsp;I should probably add an assumption that the time delay produced by the reflection act is the same &nbsp;(or negligible)&nbsp;

for all the points at the parabola internal surface.

The reflective property of the parabola has numerous practical applications. 
If a light source is  placed at the focus of a parabola the result will be a parallel beam of light directed outward along the parabola axis. 
This is how projectors, car lights, flashlights, headlights, and searchlights work.

In the present days parabolic mirrors are used at the solar energy plants to concentrate sunlight on thermal collectors containing water like vessels 
or tubes to absorb the heat energy, to heat the water and to get the water steam.

Also,  see the lessons

<A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Practical-problems-from-the-archive-related-to-ellipses-and-parabolas.lesson>Practical problems from the archive related to ellipses and parabolas</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A whispering gallery has an elliptical ceiling. &nbsp;A person standing at one focus of the ellipse can whisper and be heard

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;by another person standing at the other focus, &nbsp;because all the sound waves that reach the ceiling from one focus

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;are reflected to the other focus. &nbsp;A hall &nbsp;100 feet in length is to be designed as a whispering gallery.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the foci are located &nbsp;25 feet from the center, &nbsp;how high will the ceiling be at the center?

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;The reflector of a flashlight is in the shape of a paraboloid of revolution.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Its diameter is &nbsp;4 inches and its depth is &nbsp;1 inch.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis?

<A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/A-circle-of-maximum-radius-tangent-to-and-inscibed-in-parabola.lesson>A circle of maximum radius tangent to and inscribed in parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Find a circle of maximum radius tangent to and inscribed in parabola &nbsp;&nbsp;y = {{{4x^2}}}.

For similar lessons on ellipses see &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-ellipses.lesson>OVERVIEW of lessons on ellipses</A> &nbsp;in this site.

For similar lessons on hyperbolas see &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-hyperbolas.lesson>OVERVIEW of lessons on hyperbolas</A> &nbsp;in this site.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.