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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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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 way 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, |
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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
intersects 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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| 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:
The line 13x
-
15y
-
25 = 0 is the tangent of a hyperbola with linear eccentricity (half the focal
distance) cH =
Ö41.
Write the equation of the hyperbola.
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| Solution:
Rewrite the equation 13x
-
15y
-
25 = 0
or |
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| Using
the linear eccentricity |
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| and
the tangency condition |
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| Thus,
the equation of the hyperbola, |
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| Example:
Find the normal to the hyperbola
3x2
-
4y2 = 12 which is parallel to the line
-x +
y = 0.
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Solution:
Rewrite the equation of the hyperbola
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3x2
-
4y2 = 12
| ¸12
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The slope of the normal is equal to the slope of
the given line,
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y =
x
=>
m
= 1,
mt =
-1/mn,
so mt =
-1
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applying the tangency condition
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a2m2
-
b2 = c2
<= mt =
-1,
a2 =
4 and
b2 = 3
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4·(-1)2
-
3 = c2
=> c1,2 = ±1
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tangents, t1
::
y =
-x
+ 1 and
t2
::
y =
-x
-
1.
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The points of
tangency,
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| The
equations of the normals, |
| D1(4,
-3)
and m =
1
=>
y -
y1 = m ·(x
-x1),
y +
3 = 1·(x
- 4)
or n1
::
y = x -
7, |
| D2(-4,
3)
and m =
1 =>
y -
y1 = m ·(x
-x1),
y -
3 = 1·(x
+ 4)
or n2
::
y =
x + 7, |
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| Pre-calculus contents
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