Definition
Conic sections can be defined as the locus of point that moves so that the ratio of its distance from a fixed point called the focus to its distance from a fixed line called the directrix is constant. The constant ratio is called the eccentricity of the conic.
 

Conic Sections (Cutting Plane Passing Through a Cone)

 

Conic sections are obtained by passing a cutting plane to a right circular cone. If the cutting plane is parallel to the base of the cone (or perpendicular to the axis of the cone), a circle is defined. If the cutting plane is parallel to lateral side (or generator) of the cone, parabola is defined. For a cutting plane that is oblique to the cone (not parallel nor perpendicular to any element), ellipse is defined. For a cutting plane parallel to the axis of the cone not passing through the vertex, the section formed is hyperbola. These were characterized by the Greek mathematician Apollonius (262 B.C. – 200 B.C.).
 

Equation of Conic Sections

The equation of general conic-sections is in second-degree,
 

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.$

 

The quantity B2 - 4AC is called discriminant and its value will determine the shape of the conic.

  • If C = A and B = 0, the conic is a circle.
  • If B2 - 4AC = 0, the conic is a parabola.
  • If B2 - 4AC < 0, the conic is an ellipse.
  • If B2 - 4AC > 0, the conic is a hyperbola.

 

The product xy would have a conic with axis oblique to the coordinate axes. If B = 0, the axis of the conic is parallel to one of the coordinate axes. As of now, our concern are for conics with axis parallel to one of the coordinate axes, thus B = 0. The equation then becomes
 

$Ax^2 + Cy^2 + Dx + Ey + F = 0$.

 

Eccentricity of Conic

Eccentricity is a measure of how much a conic deviate from being circular, making the eccentricity of the circle obviously equal to zero. It is the ratio of focal distance to directrix distance of the conic section.
 

$\text{eccentricity, } e = \dfrac{\text{distance from focus to conic}}{\text{distance from conic to directrix}}$
  • If e = 0, the conic is a circle.
  • If e = 1, the conic is a parabola.
  • If e < 1, the conic is an ellipse.
  • If e > 1, the conic is a hyperbola.

 

The Circle

Definition of circle
The locus of point that moves such that its distance from a fixed point called the center is constant. The constant distance is called the radius, r of the circle.
 

General Equation of Circle (C = A)

From the general equation of conic sections, C = A. Hence, the equation of the circle is

$Ax^2 + Ay^2 + Dx + Ey + F = 0$

or

$x^2 + y^2 + Dx + Ey + F = 0$

 

Standard Equations of Circle

Circle with center at any point (h, k)
Apply Pythagorean Theorem to the figure shown below.
$(x - h)^2 + (y - k)^2 = r^2$

 

circle-01.gif

 

Circle with center at the origin
If the center is at the origin, (h, k) = (0, 0), hence,

$x^2 + y^2 = r^2$

 

The Parabola

Definition of Parabola
Parabola is the locus of point that moves such that it is always equidistant from a fixed point and a fixed line. The fixed point is called focus and the fixed line is called directrix.
 

General Equations of Parabola

From the general equation of all conic sections, either $A$ or $C$ is zero to form a parabolic section.

For $A = 0$, the equation will reduce to $Cy^2 + Dx + Ey + F = 0$ or

$y^2 + Dx + Ey + F = 0$

It is a parabola with axis horizontal, e.g., open to the right or open to the left.
 

For $C = 0$, the equation will reduce to $Ax^2 + Dx + Ey + F = 0$ or

$x^2 + Dx + Ey + F = 0$

It is a parabola with axis vertical, e.g., open upward or open downward.
 

Standard Equations of Parabola

 

Parabola with vertex at the origin and open to the right

 

From the definition
$d_1 = d_2$

${d_1}^2 = {d_2}^2$

$(x - a)^2 + (y - 0)^2 = (x + a)^2$

$(x^2 - 2ax + a^2) + y^2 = x^2 + 2ax + a^2$

$-2ax + y^2 = 2ax$

$y^2 = 4ax$

The equation we just derived was with reference to the figure shown above, thus, it is a parabola with vertex at the origin and open to the right.
 

Parabola with vertex at the origin
Parabola with vertex at the origin and open to the right.
parabola-01.gif$y^2 = 4ax$

 

Parabola with vertex at the origin and open to the left.
parabola-02.gif$y^2 = -4ax$

 

Parabola with vertex at the origin and open upward.
parabola-03.gif$x^2 = 4ay$

 

Parabola with vertex at the origin and open downward.
parabola-04.gif$x^2 = -4ay$

 

Parabola with vertex at any point (h, k)
Parabola with vertex at (h, k) and open to the right.
parabola-05.gif$(y - k)^2 = 4a(x - h)$

 

Parabola with vertex at (h, k) and open to the left.
parabola-06.gif$(y - k)^2 = -4a(x - h)$

 

Parabola with vertex at (h, k) and open upward.
parabola-07.gif$(x - h)^2 = 4a(y - k)$

 

Parabola with vertex at (h, k) and open downward.
parabola-08.gif$(x - h)^2 = -4a(y - k)$

 

Hint: To avoid memorizing the eight (8) standard equations of parabola, we will reduce it to only two (2) as follows:

$(y - k)^2 = \pm 4a(x - h)$

$(x - h)^2 = \pm 4a(y - k)$

Note that (h, k) is (0, 0) at the origin. Use positive (+) for open upward and rightward parabolas, negative (-) for open downward and leftward parabolas.
 

Elements of Parabola

  1. Focus is located at distance $a$ from vertex in the direction of parabola’s opening.
  2. Directrix is at distance $a$ from the vertex. It is a straight line located at the opposite side of parabola’s opening.
  3. Vertex is the point extremity of parabola, i.e. highest point for open downward, lowest point for open upward, rightmost point for leftward, and leftmost point for rightward. The coordinates of vertex is denoted as (h, k).
  4. Axis is the line of symmetry of parabola. It contains both the focus and the vertex and always perpendicular to the directrix.
  5. Latus Rectum, denoted by $LR$, is a line perpendicular to the axis, passing through the focus and terminates on the parabola itself. The total length of $LR$ is $4a$ ($LR = 4a$), where $a$ stands for the distance from focus to vertex.
  6. Eccentricity of parabola is always equal to 1 ($e = 1$). Thus, parabola can also be defined as a conic section of eccentricity equal to 1.

 

The Ellipse

Definition of Ellipse
Ellipse is the locus of point that moves such that the sum of its distances from two fixed points called the foci is constant. The constant sum is the length of the major axis, 2a.
 

General Equation of the Ellipse

From the general equation of all conic sections, A and C are not equal but of the same sign. Thus, the general equation of the ellipse is Ax2 + Cy2 + Dx + Ey + F = 0 or
 

$x^2 + Cy^2 + Dx + Ey + F = 0$

 

Standard Equations of Ellipse

Ellipse and all its elements

 

From the figure above,
$d_1 = \sqrt{(x + c)^2 + y^2}$   and

$d_2 = \sqrt{(x - c)^2 + y^2}$
 

From the definition above,
$d_1 + d_2 = 2a$

$\sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a$

$\sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2}$
 

Square both sides
$(x + c)^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + [ \, (x - c)^2 + y^2 \, ]$

$x^2 + 2xc + c^2 + y^2 = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} + x^2 - 2xc + c^2 + y^2$

$2xc = 4a^2 - 4a\sqrt{(x - c)^2 + y^2} - 2xc$

$4a\sqrt{(x - c)^2 + y^2} = 4a^2 - 4xc$

$a\sqrt{(x - c)^2 + y^2} = a^2 - xc$
 

Square again both sides
$a^2 [ \, (x - c)^2 + y^2 \, ] = (a^2 - xc)^2$

$a^2 [ \, x^2 - 2xc + c^2 + y^2 \, ] = a^4 - 2a^2xc + x^2c^2$

$a^2x^2 - 2a^2xc + a^2c^2 + a^2y^2 = a^4 - 2a^2xc + x^2c^2$

$a^2x^2 + a^2c^2 + a^2y^2 = a^4 + x^2c^2$

$a^2x^2 - x^2c^2 + a^2y^2 = a^4 - a^2c^2$

$(a^2 - c^2)x^2 + a^2y^2 = a^2(a^2 - c^2)$
 

From triangle OV3F2 (see figure above)
$c^2 + b^2 = a^2$

$a^2 - c^2 = b^2$
 

Thus,
$b^2x^2 + a^2y^2 = a^2b^2$
 

Divide both sides by a2b2
$\dfrac{b^2x^2}{a^2b^2} + \dfrac{a^2y^2}{a^2b^2} = \dfrac{a^2b^2}{a^2b^2}$

$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

The above equation is the standard equation of the ellipse with center at the origin and major axis on the x-axis as shown in the figure above. Below are the four standard equations of the ellipse. The first equation is the one we derived above.

Ellipse with center at the origin
Ellipse with center at the origin and major axis on the x-axis.
ellipse-01.gif$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

 

Ellipse with center at the origin and major axis on the y-axis.
ellipse-02.gif$\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1$

 

Ellipse with center at (h, k)
Ellipse with center at (h, k) and major axis parallel to the x-axis.
ellipse-03.gif$\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1$

 

Ellipse with center at (h, k) and major axis parallel to the y-axis.
ellipse-04.gif$\dfrac{(x - h)^2}{b^2} + \dfrac{(y - k)^2}{a^2} = 1$

 

Elements of Ellipse

  1. Center (h, k). At the origin, (h, k) is (0, 0).
  2. Semi-major axis = a and semi-minor axis = b.
  3. Location of foci c, with respect to the center of ellipse. $c = \sqrt{a^2 - b^2}$.
  4. Length latus rectum, LR
    semi-latus rectum of ellipseConsider the right triangle F1QF2:
  5. Based on the definition of ellipse
    $z + \frac{1}{2}LR = 2a$

    $z = 2a - \frac{1}{2}LR$

    $z = \dfrac{4a - LR}{2}$
     

    By Pythagorean Theorem
    $(2c)^2 + (\frac{1}{2}LR)^2 = z^2$

    $4c^2 + \frac{1}{4}(LR)^2 = \left( \dfrac{4a - LR}{2} \right)^2$

    $4c^2 + \frac{1}{4}(LR)^2 = \dfrac{(4a - LR)^2}{4}$

    $16c^2 + (LR)^2 = (4a - LR)^2$

    $16c^2 + (LR)^2 = 16a^2 - 8a(LR) + (LR)^2$

    $16c^2 = 16a^2 - 8a(LR)$

    $8a(LR) = 16a^2 - 16c^2$

    $LR = \dfrac{16a^2 - 16c^2}{8a}$

    $LR = \dfrac{16(a^2 - c^2)}{8a}$

    $LR = \dfrac{2b^2}{a}$

    You can also find the same formula for the length of latus rectum of ellipse by using the definition of eccentricity.
     

  6. Eccentricity, e
    $e = \dfrac{\text{distance from focus to ellipse}}{\text{distance from ellipse to directrix}}$
     

    From the figure of the ellipse above,
    $e = \dfrac{d_3}{d_4} = \dfrac{a}{d} = \dfrac{a - c}{d - a}$

    From
    $\dfrac{a}{d} = \dfrac{a - c}{d - a}$

    $ad - a^2 = ad - cd$

    $d = a^2 / c$
     

    Thus,
    $e = \dfrac{a}{d} = \dfrac{a}{a^2 / c}$

    $e = \dfrac{c}{a} \lt 1.0$

     

  7. Location of directrix d, with respect to the center of ellipse.

    From the derivation of eccentricity,

    $d = \dfrac{a}{e} \, \text{ or } d = \dfrac{a^2}{c}$

 

The Hyperbola

Definition
Hyperbola can be defined as the locus of point that moves such that the difference of its distances from two fixed points called the foci is constant. The constant difference is the length of the transverse axis, 2a.
 

General Equation of Hyperbola

From the general equation of any conic (A and C have opposite sign, and can be A > C, A = C, or A < C.)

$Ax^2 - Cy^2 + Dx + Ey + F = 0 \,$ or
 

$\, x^2 - Cy^2 + Dx + Ey + F = 0$

 

Standard Equations of Hyperbola

Elements of Hyperbola

 

From the definition:
$d_2 - d_1 = 2a$

$\sqrt{(x + c)^2 + (y - 0)^2} - \sqrt{(x - c)^2 + (y - 0)^2} = 2a$

$\sqrt{(x + c)^2 + y^2} = 2a + \sqrt{(x - c)^2 + y^2}$

$(x + c)^2 + y^2 = 4a^2 + 4a\sqrt{(x - c)^2 + y^2} + [ \, (x - c)^2 + y^2 \, ]$

$(x + c)^2 = 4a^2 + 4a\sqrt{(x - c)^2 + y^2} + (x - c)^2$

$x^2 + 2xc + c^2 = 4a^2 + 4a\sqrt{(x - c)^2 + y^2} + x^2 - 2cx + c^2$

$4xc - 4a^2 = 4a\sqrt{(x - c)^2 + y^2}$

$xc - a^2 = a\sqrt{(x - c)^2 + y^2}$

$(xc - a^2)^2 = a^2 [ \, (x - c)^2 + y^2 \, ]$

$x^2c^2 - 2xca^2 + a^4 = a^2 [ \, (x - c)^2 + y^2 \, ]$

$x^2c^2 - 2xca^2 + a^4 = a^2 [ \, x^2 - 2xc + c^2 + y^2 \, ]$

$x^2c^2 - 2xca^2 + a^4 = a^2x^2 - 2xca^2 + a^2c^2 + a^2y^2$

$x^2c^2 + a^4 = a^2x^2 + a^2c^2 + a^2y^2$

$(x^2c^2 - a^2x^2) - a^2y^2 = a^2c^2 - a^4$

$(c^2 - a^2)x^2 - a^2y^2 = a^2(c^2 - a^2)$
 

From the figure:
$c^2 = a^2 + b^2$

$c^2 - a^2 = b^2$
 

Thus,
$b^2x^2 - a^2y^2 = a^2b^2$

$\dfrac{b^2x^2}{a^2b^2} - \dfrac{a^2y^2}{a^2b^2} = \dfrac{a^2b^2}{a^2b^2}$

$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$

The equation we just derived above is the standard equation of hyperbola with center at the origin and transverse axis on the x-axis (see figure above). Below are the four standard equations of hyperbola. The first equation is the one we derived just derived.
 

Hyperbola with center at the origin
Hyperbola with center at the origin and transverse axis on the x-axis.
hyperbola-02.gif$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$

 

Hyperbola with center at the origin and transverse axis on the y-axis.
hyperbola-03.gif$\dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1$

 

Hyperbola with center at any point (h, k)
Hyperbola with center at (h, k) and transverse axis parallel to the x-axis.
hyperbola-04.gif$\dfrac{(x - h)^2}{a^2} - \dfrac{(y - k)^2}{b^2} = 1$

 

Hyperbola with center at (h, k) and transverse axis parallel to the y-axis.
hyperbola-05.gif$\dfrac{(y - k)^2}{a^2} - \dfrac{(x - h)^2}{b^2} = 1$

 

Elements of Hyperbola

  1. Center (h, k). At the origin, (h, k) is (0, 0).
  2. Transverse axis = 2a and conjugate axis = 2b
  3. Location of foci c, relative to the center of hyperbola.
     
    $c = \sqrt{a^2 + b^2}$

     

  4. Latus rectum, LR
     
    $LR = \dfrac{2b^2}{a}$

     

  5. Eccentricity, e
    The eccentricity of hyperbola is always greater than one.
     
    $e = \dfrac{c}{a} > 1.0$

     

  6. Location of directrix d relative to the center of hyperbola.
     
    $d = \dfrac{a}{e}$   or   $d = \dfrac{a^2}{c}$

     

  7. Equation of asymptotes.
     
    $y - k = \pm m(x - h)$

     
    where
    m is (+) for upward asymptote and m is (-) for downward.
    m = b/a if the transverse axis is horizontal and
    m = a/b if the transverse axis is vertical

 

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