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Plane Analytical Geometry

Equation of Parabola and Their Tangent Line

By:

6th

Group

Members:

1. Deni Hariyanto ( E1R0110 )2. M. Arif Hidayatullah ( E1R0110 )3. Niswah Qurrota Ayuni ( E1R011035 )

Program Studi Pendidikan Matematika

Fakultas Keguruan dan Ilmu Pendidikan

Universitas Mataram

2011-2012


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1. Definition of a ParabolaIn the following graph,

The conic is the locus of a point which moves so that the ratio of its distance from a fixed point (Focus ) and a fixed line is constant.

The directrix is the liney = p. The focal distance is |p| (Distance from the origin to the focus, and from the origin to the

directrix. We take absolute value because distance is positive.)

The point P (x,y) represents any moving point on the curve. dis the distance from any point P (x,y) to the Focus (0,p) in the parabola case its the same as

the distance from P (x,y) to the directrix.

The eccentricity (e) is the constant ratio d : d The axis of parabola is the line through Fperpendicular to the directrix. The vertex is the point O where the parabola cuts its axis

Classes of Conics:

The form of a conic depends on the value of the ratio d : d Ife = 1, that is if d = d, the conic is called a parabola

If e < 1, the conic is called an ellipse

If e > 1, the conic is called a hyperbola.

So the the parabola is defined as the locus of a point which moves so that it is always the same

distance from a fixed point (called the focus) and a given line (called the directrix).

2. The Equation of Parabola

By the definition of parabola it follows that a locus is a paraola if it satisfies the condition FP=

QP.

P

Q

P

d

d


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Adding to our diagram from above, we see that the distance d = QP. Now, using theDistance

Formulaon the general points F (0,p) and P(x,y), and equating it to our value QP= d=y+p, we

have:

FP = QP

Squaring both sides gives:

Simplifying gives us the formula for a parabola:

In more familiar form, with "y = " on the left, we can write this as:

wherep is the focal distance of the parabola.

For reference we list the two forms for equations of parabolas to standard position:

3. The Formula for a Parabola - Vertical AxisNow if we assume that the axis of the parabola is the y axis, so

that its focus is a point (0,p) and its direcrtix is the line y = -p,

then a similiar derivation ( withx andy interchanged ) yields the

equation

For the parabola. In this case the parabola is

Concave upward if p > 0 Concave downward if p < 0

2= 4

Focus is ( 0, p ) Directrix is y = -p Symmetry with respect to the y axis

2= 4

Focus is ( p, 0 ) Directrix is x = -p Symmetry with respect to the x axis

As we known the formula to

find the distance dbetween

two points

P1 (x1, y1) and P2 (x2, y2)
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If || is large, the focus and directrix are far apart and the parabola looks wide and blunt. If |p| issmall, the focus and directrix are close together and the parabola looks long and slender.

Example - Parabola with Vertical Axis

Sketch the parabola

Find the focal length and indicate the focus and the directrix on your graph.

The focal length is found by equating the general expression fory

and our particular example:

So we have:

This givesp = 0.5

So the focus will be at (0,0.5) and the directrix is

the liney = 0.5.

Our curve is as follows:

Note: Even though the sides look as though theybecome straight asx increases, in fact they do not.

The sides of a parabola just get steeper and steeper

(but are never vertical, either).

4. Parabolas with Horizontal AxisIf we assume that the axis of the parabola is the x axis, so thatits focus is a point (p,0) and its direcrtix is the line x = -p, then

a similiar derivation ( withx andy interchanged ) yields the

equation

For the parabola. In this case the parabola is

Opening to the right if p > 0 Opening to the left if p < 0

Answer
http://www.intmath.com/plane-analytic-geometry/ans-4.php?a=0http://www.intmath.com/plane-analytic-geometry/ans-4.php?a=0http://www.intmath.com/plane-analytic-geometry/ans-4.php?a=0


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If || is large, the focus and directrix are far apart and the parabola looks wide and blunt. If |p| issmall, the focus and directrix are close together and the parabola looks long and slender.

Example - Parabola with Horizontal Axis

Sketch the curve and find the equation of the parabola with focus (-2,0) and directrixx = 2.

In this case, we have the following graph:

After sketching, we can see that the equation required is in the following form, since we have a

horizontal axis:

Sincep=2 (from the question), we can directly write the equation of the parabola:

5. Shifting the Vertex of a Parabola from the OriginThis is a similar concept to the case when we shifted the centre of acirclefrom the origin.

To shift the vertex of a parabola from (0, 0) to (h, k), eachx in the equation becomes (x h)

and eachy becomes (y k).

So if the axis of a parabola is vertical, and the vertex is at (h, k), from

, we have

If the axis of a parabola is horizontal, and the vertex is at (h, k), from the equationbecomes

Answer
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6. Tangent Line of ParabolaTangent. We have to hitherto regarded a tangent to a circle as a line which touches the circle in one

point and only one. This definition of tangent does not, however, apply to curves in general.

Thus, we may say that lis tangent to this curve atA, altough lcuts the curve again atB.

Ifc is any curve andPis any point on it, we define the tangent to c atPas follows:

Take another point Q upon the curve, and draw the secantPQ. Letting Q move along toward the

P, as Q approachesPthe secantPQ turns aboutPand approaches a definite limiting positionPT.

The LinePTis then said to be tangent to the curve c atP.

The secantPQ cuts the curve in two pointsPand Q. As the secant approaches the position of the

tangentPT, the point Q approaches the pointP; and when the secant coincides with the tangent, Q

coincides withP.

We therefore say that a tangent to a curve cuts the curve in two coincident points at the points of

tangency.

a. Slope of a curve.The slope of the tangent at a pointPon a curve is called also the slope of the curve at the point P.

To find the slope of parabol a at any poin t on the parabola


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Solution:Take another point Q(x1+h, y1+k) on the parabola. Then the slope of secantP1Q is, and

the slope m of the tangent atP1 is the limit of the slope of the secant as Q approachesP1, That is,

SinceP1 and Q are both on the parabola, we have

(1) (2)Subtracting from (1) and (2), we have

;Whence

Now when and hence

. But lim , the slope of the tangent

Hence

That is the slope of parabola at any point is equal to 2p divided by the ordinate of thepoint.b. Tangent to the Parabola

To find the equation of the tangent to the parabola a at any point on theparabola.

Solution:Since the tangent passes through the poin t and has the slope , its equationis

Clearing of fraction, we have

And since is on the parabola a, we have , and the equation takesthe convenient form


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Example

Find the equation of the tangent to the parabol a at the point

Equation of tangent

(

) .

Answer
http://www.intmath.com/plane-analytic-geometry/ans-4.php?a=0http://www.intmath.com/plane-analytic-geometry/ans-4.php?a=0http://www.intmath.com/plane-analytic-geometry/ans-4.php?a=0


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BOOK REFERENCE

Ellis, Robert and Denny Gulick. 1982. Calculus With Alaytic Geometry. Harcourt Brace Jovanovich

Publishers. New YorkUSA.

Siceloff, Lewis Parker, etc. 1922.Analytic Geometry. Ginn and Company. Boston - USA.

http://www.intmath.com/plane-analytic-geometry/4-parabola.php
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