Common Tangent Between Hyperbola and Circle

Understanding the concept of a common tangent between a hyperbola and a circle is an important topic in coordinate geometry. A common tangent is a line that touches both the hyperbola and the circle at distinct points, without crossing either of them. This topic is particularly relevant for students preparing for exams that include geometry or analytic geometry.

Hyperbola

A hyperbola is a type of conic section that can be defined as the locus of all points in a plane such that the difference of the distances from two fixed points (foci) is constant. The standard equation of a hyperbola centered at the origin with its transverse axis along the x-axis is:

$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$

where $a$ is the distance from the center to a vertex on the x-axis, and $b$ is the distance from the center to a vertex on the conjugate axis.

Circle

A circle is a simple closed shape in a plane that consists of all points in that plane that are at a given distance (radius) from a given point (the center). The standard equation of a circle with center at the origin and radius $r$ is:

$$ x^2 + y^2 = r^2 $$

Finding the Common Tangent

To find the common tangent between a hyperbola and a circle, we need to find a line that satisfies the tangency condition for both curves. The general equation of a line in the plane is:

$$ y = mx + c $$

where $m$ is the slope and $c$ is the y-intercept.

For the Hyperbola

The condition for tangency to the hyperbola is that the line must satisfy the equation of the hyperbola and the discriminant of the resulting quadratic equation must be zero. Substituting the equation of the line into the hyperbola's equation, we get:

$$ \frac{(mx + c)^2}{a^2} - \frac{y^2}{b^2} = 1 $$

Solving for $y$ and setting the discriminant to zero will give us the condition for $m$ and $c$.

For the Circle

Similarly, for the circle, the condition for tangency is that the line must satisfy the circle's equation and the discriminant of the resulting quadratic equation must be zero. Substituting the line's equation into the circle's equation, we get:

$$ (mx + c)^2 + y^2 = r^2 $$

Again, solving for $y$ and setting the discriminant to zero will give us the condition for $m$ and $c$.

Table of Differences and Important Points

Feature	Hyperbola	Circle
Equation	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$x^2 + y^2 = r^2$
Tangency Condition	Discriminant of quadratic in $y$ must be zero	Discriminant of quadratic in $y$ must be zero
Line Equation	$y = mx + c$	$y = mx + c$
Determining $m$ and $c$	Solve for $y$ in hyperbola equation and set discriminant to zero	Solve for $y$ in circle equation and set discriminant to zero

Examples

Example 1: Hyperbola Tangent

Given the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, find the equation of the tangent line with slope $m = 2$.

Solution:

The equation of the tangent line is $y = 2x + c$. Substituting into the hyperbola's equation:

$$ \frac{(2x + c)^2}{9} - \frac{y^2}{16} = 1 $$

Solving for $y$ and setting the discriminant to zero, we find the value of $c$ that satisfies the tangency condition.

Example 2: Circle Tangent

Given the circle $x^2 + y^2 = 25$, find the equation of the tangent line with slope $m = -1$.

Solution:

The equation of the tangent line is $y = -x + c$. Substituting into the circle's equation:

$$ (-x + c)^2 + y^2 = 25 $$

Solving for $y$ and setting the discriminant to zero, we find the value of $c$ that satisfies the tangency condition.

Conclusion

Finding a common tangent between a hyperbola and a circle involves using the general equation of a line and applying the tangency condition to both the hyperbola and the circle. By setting the discriminant of the resulting quadratic equations to zero, we can determine the slope and y-intercept of the tangent line. This process requires a good understanding of algebra and the properties of conic sections.