
Conic
Sections 


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

Polar and pole of 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 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 

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. 

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.



Polar and pole of the hyperbola

If from a point
A(x_{0}, y_{0}), 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, 

b^{2}x_{0}x

a^{2}y_{0}y
= a^{2}b^{2} 
the equation of the
polar of the point A(x_{0}, y_{0}). 


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:
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. 
Solution:
Find
the equation of the ellipse by solving the system of
equations, 


thus,
the equation of the ellipse 


The equation of the hyperbola by solving the system of
equations,





Therefore, equation of the hyperbola 


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)
4x^{2}
+ 9y^{2} = 36 (2)
=> (1)
4(4y^{2}
+ 4) + 9y^{2} = 36,
25y^{2} = 20,
y_{1,2} = ±2/Ö5,
x_{1,2} = ±6/Ö5 
(2) x^{2}

4y^{2} = 4
=>
x^{2} = 4y^{2}
+ 4 
Tangent
of the ellipse and the hyperbola at the intersection S_{1}(6/Ö5,
2/Ö5), 
S_{1}(6/Ö5,
2/Ö5)
=> t_{e
}::
b^{2}x_{1}x
+ a^{2}y_{1}y
= a^{2}b^{2},
4x + 3y
= 6/Ö5
or y =
(4/3)x
+ 2Ö5, 
S_{1}(6/Ö5,
2/Ö5)
=>
t_{h}_{
}::
b^{2}x_{1}x

a^{2}y_{1}y
= a^{2}b^{2},
3x 
4y = 2/Ö5
or y
= (3/4)x 
Ö5/2. 

fulfilled is the perpendicularity condition.
Therefore, the angle between curves j
= 90°. 








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