Chapter 9, Problem 10STP

To determine

To find: The maximum number of solutions of equations.

Answer to Problem 10STP

The maximum number of solutions is 4.

Explanation of Solution

Given information:

The given information is a equation which consists of a circle and a hyperbola.

Calculation:

When a quadratic system contains two conic sections, the system may have 0, 1, 2, 3 or 4 solutions.

If the hyperbola and circle have same center and radius of the circle is greater than the half of the length of transverse axis of hyperbola, system consisting of two conics will have four solutions.

For example, the equation of the circle is,

  $x^2+y^2=\mathrm{36......}\left(1\right)$

The circle has its center at origin and radius equal to $6$ units.

Compute a hyperbola with center at origin and transeverse axis of length less than $12$ units.

For example,

  $\frac{x^2}{9}-\frac{y^2}{4}=\mathrm{1........}\left(2\right)$

The above hyperbola intersects the circle $\left(1\right)x^2+y^2=36$ at four points

  $\left(5.28,2.9\right),\left(-5.28,2.9\right),\left(5.28,-2.9\right),\left(-5.28,-2.9\right)$

The graph of the two conics is shown in figure $\left(1\right)$

  

Figure $\left(1\right)$