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

Analytic geometry - Conic sections

Hyperbola and line

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 b²

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, to examine conditions which determine position of a line in relation to a hyperbola we solve the system of equations,

y = mx + c then if, a²m² - b² > c² the line intersects the hyperbola at two points,

b²x² - a²y² = a²b²a²m² - b² < c² the line and the hyperbola do not intersect.

a²m² - b² = c² the line is the tangent of the hyperbola,

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

A line is the tangent to the hyperbola if a²m² - b² = c², this equality is called the tangency condition.

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.

The tangency condition also states 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 intersects the hyperbola.

The equation of the tangent at the point on the hyperbola

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,

the tangency point or the point of contact.

Thus, the intercept and slope of the tangent

or b²x₁x - a²y₁y = a²b² the equation of the tangent at a point P₁(x₁, y₁) on the hyperbola.

Polar and pole of the hyperbola

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 = a²b²the equation of the polar of the point A(x₀, y₀).

Construction of the tangent at the point on the hyperbola

The tangent at the point P₁(x₁, y₁) on the hyperbola is the bisector of the angle F₁P₁F₂ subtended by

focal radii, r₁ and r₂ at P₁.

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

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 F₂, and from F₁as center, draw an arc of radius 2a.

These arcs intersect at points S₁ and S₂.

Tangents are the perpendicular bisectors of the line segments F₂S₁ and F₂S₂.

Tangents can also be drawn as lines through A and the intersection points of lines through F₁S₁ and F₁S₂, with the hyperbola.

These intersections are at the same time the points of contact D₁ and D₂.

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.