Applications of Polynomial Equation in Geometry

Applications of Polynomial Equation in Geometry

Certain geometrical properties are proved using polynomial equations. We discuss a few geometric properties here.

Example 3.14

Prove that a line cannot intersect a circle at more than two points.

Solution

By choosing the coordinate axes suitably, we take the equation of the circle as x² + y² = r ² and the equation of the straight line as y = mx + c . We know that the points of intersections of the circle and the straight line are the points which satisfy the simultaneous equations

x²  + y²   = r²                  ... (1)

y = mx + c       ... (2)

If we substitute mx + c for y in (1), we get

x² + (mx + c)² - r² = 0

which is same as the quadratic equation

(1+ m²)x² + 2mcx + (c² - r²) = 0 .        ... (3)

This equation cannot have more than two solutions, and hence a line and a circle cannot intersect at more than two points.

It is interesting to note that a substitution makes the problem of solving a system of two equations in two variables into a problem of solving a quadratic equation.

Further we note that as the coefficients of the reduced quadratic polynomial are real, either both roots are real or both imaginary. If both roots are imaginary numbers, we conclude that the circle and the straight line do not intersect. In the case of real roots, either they are distinct or multiple roots of the polynomial. If they are distinct, substituting in (2), we get two values for y and hence two points of intersection. If we have equal roots, we say the straight line touches the circle as a tangent. As the polynomial (3) cannot have only one simple real root, a line cannot cut a circle at only one point.

Note

A technique similar to the one used in example 3.14 may be adopted to prove

• two circles cannot intersect at more than two points.

• a circle and an ellipse cannot intersect at more than four points.