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| Conic
Sections |
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Hyperbola
and Line
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 Hyperbola and line relationships
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Condition for a line to be the tangent to the hyperbola -
tangency condition |
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The equation of the tangent at the point on the hyperbola
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Polar and pole of the hyperbola
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Construction of the tangent at the point on the hyperbola
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Construction of tangents from a point outside the hyperbola
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Hyperbola and
line examples
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| Hyperbola and line relationships
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| Let examine relationships between a hyperbola and a line passing through the center of the hyperbola, i.e., the
origin. A line y =
mx intersects the hyperbola at two
points if
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the slope |m|
<
b/a but
if |m|
> b/a
then,
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the line y =
mx does not intersect the hyperbola at all.
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| The diameters of a hyperbola are straight lines
passing through its center.
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The asymptotes divide these two pencils of diameters
into, one which intersects the curve at two points, and
the
other which do not intersect.
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A diameter of a conic section is a line which passes
through the midpoints of parallel chords.
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Conjugate diameters of the hyperbola (or the ellipse)
are two diameters such that each bisects all chords
drawn parallel to the other.
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| As the equation of a hyperbola can be obtained from the equation of an ellipse by changing the sign of
b2
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that is, a2(1
-
e2)
= -a2(e2
-
1) =
-b2, |
| this way, we can use other formulas relating to the ellipse to obtain corresponding
formulas for the hyperbola. |
| Therefore, when we examine conditions which determine position of a line in relation to a hyperbola that is, |
| when
solve the system of
equations,
y =
mx + c |
| b2x2
-
a2y2
= a2b2
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then if, a2m2
-
b2
> c2
the line intersects the hyperbola at two points,
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a2m2
-
b2 = c2
the line is the tangent of the hyperbola,
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a2m2
-
b2
< c2
the line and the hyperbola do not intersect. |
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| Condition for a line to be the tangent to the hyperbola -
tangency condition
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| A line is the tangent to the hyperbola if
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a2m2
-
b2 = c2. |
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| Regarding
the asymptotes, to which c
= 0, this condition gives
|m|
= b/a
and that is why we can say that the hyperbola touches the asymptotes at infinity. |
| From the tangency condition it also follows that the slopes of the tangents
will satisfy the condition if |
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| That is, the tangents to the hyperbola can only be parallel to a line belonging to the pencil of lines that do not
intersect the hyperbola. |
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| The equation of the tangent at the point on the hyperbola
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As we already mentioned, the points of contact of a line and the hyperbola can be obtained from the
corresponding formula for the ellipse by changing b2
with -b2
thus |
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the
tangency point or the point of contact. |
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| So, the intercept and slope of the
tangent |
 |
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| or |
b2x1x
-
a2y1y
= a2b2 |
the equation of the tangent at a point
P1(x1,
y1)
on the hyperbola.
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| Polar and pole of the hyperbola
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If from a point
A(x0, y0), exterior to the hyperbola, drawn are tangents, then the secant line passing through
the contact points, is the polar of the point A.
The point A
is called the pole of the polar. |
| The equation of the
polar is derived the same way as for the ellipse, |
| |
b2x0x
-
a2y0y
= a2b2 |
the equation of the
polar of the point A(x0, y0). |
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| Construction of the tangent at the point on the hyperbola
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| The tangent at the point
P1(x1,
y1)
on the hyperbola is the bisector of the angle F1P1F2
subtended by focal |
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radii,
r1
and
r2 at
P1 .
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The proof shown for the ellipse applied to
the hyperbola gives,
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| or |
 |
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See the title
'
The angle between the focal radii at a point of the
ellipse'.
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| Construction of tangents from a point outside the hyperbola |
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With A as center draw an arc through
F2,
and from F1as center, draw an arc of radius
2a.
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These arcs intersect at points
S1
and
S2.
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Tangents are the
perpendicular bisectors of the line segments F2S1
and
F2S2.
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Tangents can also be drawn as lines through
A and
the intersection points of lines through F1S1
and
F1S2,
with the hyperbola.
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These intersections are at the same
time the points of contact
D1
and
D2.
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| Hyperbola and line examples
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| Example:
Determine the
semi-axis
a
such that the line
5x
-
4y
-
16 = 0 be the tangent of the hyperbola |
| 9x2
-
a2y2 = 9a2. |
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Solution:
Rewrite the equation
9x2
-
a2y2 = 9a2 | ¸
9a2
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| and
the equation of the tangent 5x
-
4y
-
16 = 0
or |
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| Then,
plug the slope and the intercept into tangency condition,
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| Therefore,
the given line is the tangent of the hyperbola |
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| Example:
Find the angle
between the ellipse, which passes through points, A(Ö5,
4/3) and B(1,
4Ö2/3), and
the hyperbola whose asymptotes are y
= ±
x/ 2
and the linear eccentricity or half of the focal distance |
| c
= Ö5. |
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Solution:
Find
the equation of the ellipse by solving the system of
equations, |
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| thus,
the equation of the ellipse |
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The equation of the hyperbola by solving the system of
equations,
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| Therefore, equation of the hyperbola |
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| Angle between curves is the angle between tangents at intersection of the curves. By solving the system of
equations of the curves we obtain the points of intersection, |
| (1)
4x2
+ 9y2 = 36 (2)
=> (1)
4(4y2
+ 4) + 9y2 = 36,
25y2 = 20,
y1,2 = ±2/Ö5,
x1,2 = ±6/Ö5 |
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(2) x2
-
4y2 = 4
=>
x2 = 4y2
+ 4 |
| Tangent
of the ellipse and the hyperbola at the intersection S1(6/Ö5,
2/Ö5), |
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S1(6/Ö5,
2/Ö5)
=> te
::
b2x1x
+ a2y1y
= a2b2,
4x + 3y
= 6/Ö5
or y =
(-4/3)x
+ 2Ö5, |
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S1(6/Ö5,
2/Ö5)
=>
th
::
b2x1x
-
a2y1y
= a2b2,
3x -
4y = 2/Ö5
or y
= (3/4)x -
Ö5/2. |
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| fulfilled is the perpendicularity condition.
Therefore, the angle between curves j
= 90°. |
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| Conic
sections contents |
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