Point of Intersection of Rectangular Hyperbola and a Circle

Understanding the point of intersection between a rectangular hyperbola and a circle involves solving a system of equations that represent each curve. A rectangular hyperbola is a specific type of hyperbola where the asymptotes are perpendicular to each other, and a circle is a set of points that are equidistant from a fixed point called the center.

Rectangular Hyperbola

A rectangular hyperbola can be represented by the standard equation:

$$ xy = c^2 $$

where ( c ) is a constant. This equation is derived from the general hyperbola equation ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ) with ( a = b ) and a rotation of 45 degrees.

Circle

A circle can be represented by the standard equation:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

where ( (h, k) ) is the center of the circle and ( r ) is the radius.

Finding the Points of Intersection

To find the points of intersection, we need to solve the system of equations formed by the equations of the hyperbola and the circle. This typically involves substituting one equation into the other and solving for the variables.

Example

Let's consider a rectangular hyperbola ( xy = 4 ) and a circle ( (x - 1)^2 + (y - 2)^2 = 5 ).

Step 1: Substitute Hyperbola Equation into Circle Equation

From the hyperbola equation, we have ( y = \frac{4}{x} ). Substituting this into the circle equation:

$$ (x - 1)^2 + \left(\frac{4}{x} - 2\right)^2 = 5 $$

Step 2: Solve for ( x )

Expanding and simplifying the equation, we get a quadratic equation in terms of ( x ). Solve this equation to find the ( x )-coordinates of the intersection points.

Step 3: Find Corresponding ( y )-Coordinates

Using the ( x )-coordinates, substitute back into ( y = \frac{4}{x} ) to find the corresponding ( y )-coordinates.

Step 4: Verify the Points

Verify that the points satisfy both the hyperbola and the circle equations.

Differences and Important Points

Aspect	Rectangular Hyperbola	Circle
Standard Equation	( xy = c^2 )	( (x - h)^2 + (y - k)^2 = r^2 )
Symmetry	Central symmetry	Radial symmetry
Asymptotes	Perpendicular lines	None
Intersection with Axes	None	( x = \pm r + h ), ( y = \pm r + k )
Definition	Product of distances from the axes is constant	Distance from center is constant

Formulas

• Rectangular Hyperbola: ( xy = c^2 )
• Circle: ( (x - h)^2 + (y - k)^2 = r^2 )
• Intersection Points: Solve the system of equations formed by the two curves.

Example to Explain Important Points

Consider a rectangular hyperbola ( xy = 1 ) and a circle ( x^2 + y^2 = 4 ).

Step 1: Substitute Hyperbola Equation into Circle Equation

From the hyperbola equation, ( y = \frac{1}{x} ). Substituting into the circle equation:

$$ x^2 + \left(\frac{1}{x}\right)^2 = 4 $$

Step 2: Solve for ( x )

Multiplying through by ( x^2 ) to clear the fraction:

$$ x^4 + 1 = 4x^2 $$

Rearrange to form a quadratic in ( x^2 ):

$$ x^4 - 4x^2 + 1 = 0 $$

Solving this quadratic gives us ( x^2 = 2 \pm \sqrt{3} ). Taking square roots gives us the ( x )-coordinates:

$$ x = \pm \sqrt{2 + \sqrt{3}} \quad \text{or} \quad x = \pm \sqrt{2 - \sqrt{3}} $$

Step 3: Find Corresponding ( y )-Coordinates

Substitute the ( x )-values back into ( y = \frac{1}{x} ) to find the corresponding ( y )-values.

Step 4: Verify the Points

Check that these points satisfy both the hyperbola and the circle equations.

Through this process, we can find the exact points where a rectangular hyperbola intersects with a circle. The solution involves algebraic manipulation and solving quadratic equations. The intersection points can provide important geometric insights into the relationship between the two curves.