

Hyperbola
and Line

Hyperbola and line relationships

Condition for a line to be the tangent to the hyperbola 
tangency condition 
The equation of the tangent at the point on the hyperbola

Construction of the tangent at the point on the hyperbola

Construction of tangents from a point outside the hyperbola

Hyperbola and line examples






Hyperbola and line relationships

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

the slope m
<
b/a but
if m
> b/a
then,

the line y =
mx does not intersect the hyperbola at all.

The diameters of a hyperbola are straight lines
passing through its center.

The asymptotes divide these two pencils of diameters
into, one which intersects the curve at two points, and
the
other which do not intersect.

A diameter of a conic section is a line which passes
through the midpoints of parallel chords.

Conjugate diameters of the hyperbola (or the ellipse)
are two diameters such that each bisects all chords drawn parallel to the other.




As the equation of a hyperbola can be obtained from the equation of an ellipse by changing the sign of
b^{2}

that is, a^{2}(1

e^{2})
= a^{2}(e^{2}

1) =
b^{2}, 
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 
b^{2}x^{2}

a^{2}y^{2}
= a^{2}b^{2}


then if, a^{2}m^{2}

b^{2}
> c^{2}
the line intersects the hyperbola at two points,

a^{2}m^{2}

b^{2} = c^{2}
the line is the tangent of the hyperbola,

a^{2}m^{2}

b^{2}
< c^{2}
the line and the hyperbola do not intersect. 

Condition for a line to be the tangent to the hyperbola 
tangency condition

A line is the tangent to the hyperbola if

a^{2}m^{2}

b^{2} = c^{2}. 


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, 

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. 

The equation of the tangent at the point on the hyperbola

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 b^{2}
with b^{2}
thus 


the
tangency point or the point of contact. 

So, the intercept and slope of the
tangent 


or 
b^{2}x_{1}x

a^{2}y_{1}y
= a^{2}b^{2} 
the equation of the tangent at a point
P_{1}(x_{1},
y_{1})
on the hyperbola.



Construction of the tangent at the point on the hyperbola

The tangent at the point
P_{1}(x_{1},
y_{1})
on the hyperbola is the bisector of the angle F_{1}P_{1}F_{2}
subtended by focal 
radii,
r_{1}
and
r_{2} at
P_{1} .

The proof shown for the ellipse applied to
the hyperbola gives,


or 



See the title
'
The angle between the focal radii at a point of the
ellipse'.





Construction of tangents from a point outside the hyperbola 
With A as center draw an arc through
F_{2},
and from F_{1}as center, draw an arc of radius
2a.

These arcs intersect at points
S_{1}
and
S_{2}.

Tangents are the
perpendicular bisectors of the line segments F_{2}S_{1}
and
F_{2}S_{2}.

Tangents can also be drawn as lines through
A and
the intersection points of lines through F_{1}S_{1}
and F_{1}S_{2},
with the hyperbola.

These intersections are at the same
time the points of contact
D_{1}
and
D_{2}.





Hyperbola and line examples

Example:
Determine the semiaxis
a
such that the line
5x

4y

16 = 0 be the tangent of the hyperbola 
9x^{2}

a^{2}y^{2} = 9a^{2}. 
Solution:
Rewrite the equation
9x^{2}

a^{2}y^{2} = 9a^{2}  ¸
9a^{2}


and the equation of the tangent
5x

4y

16 = 0
or 


Then,
plug the slope and the intercept into tangency condition,


Therefore,
the given line is the tangent of the hyperbola 



Example:
The line 13x

15y

25 = 0 is the tangent of a hyperbola with linear eccentricity (half the focal
distance) c_{H} =
Ö41.
Write the equation of the hyperbola.

Solution:
Rewrite the equation 13x

15y

25 = 0
or 


Using
the linear eccentricity 


and
the tangency condition 

Thus,
the equation of the hyperbola, 



Example:
Find the normal to the hyperbola
3x^{2}

4y^{2} = 12 which is parallel to the line
x +
y = 0.

Solution:
Rewrite the equation of the hyperbola

3x^{2}

4y^{2} = 12
 ¸12


The slope of the normal is equal to the slope of
the given line,

y =
x
=>
m
= 1,
m_{t} =
1/m_{n},
so m_{t} =
1

applying the tangency condition

a^{2}m^{2}

b^{2} = c^{2}
<= m_{t} =
1,
a^{2} =
4 and
b^{2} = 3

4·(1)^{2}

3 = c^{2}
=> c_{1,2} = ±1

tangents, t_{1
}::
y =
x
+ 1 and
t_{2
}::
y =
x

1.

The points of
tangency,





The
equations of the normals, 
D_{1}(4,
3)
and m =
1
=>
y 
y_{1} = m ·(x
x_{1}),
y +
3 = 1·(x
 4)
or n_{1}_{
}::
y = x 
7, 
D_{2}(4,
3)
and m =
1 =>
y 
y_{1} = m ·(x
x_{1}),
y 
3 = 1·(x
+ 4)
or n_{2}_{
}::
y =
x + 7, 








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