Chapter 7.1, Problem 94E

To determine

To solve the equations of lines, where $y=x^2$ intersect the equations of lines at two points, one point and no points.

Answer to Problem 94E

The intersection of parabola and line depends on the nature of roots of the polynomial of degree 2.

Explanation of Solution

Given information: The equation of parabola is: $y=x^2$

Let us the equation of line is: $y=mx+c$

Concept used: To get the point of intersection of the line and the parabola, substitute $y=mx+c$ in the equation of parabola $y=x^2$ . Different cases arises as:

1. If the roots of the equation are real and distinct, then Lines intersect at two points.
2. If the roots of the equation are real and equal, then Lines intersect at onepoints.
3. If the roots of the equation are imaginary, then Lines never intersect.

Calculation:

  $y=x^2(1)$

  $y=mx+c(2)$

Put the value $y=mx+c$ in eq. (1).

The eq. (1) becomes as:

  $mx+c=x^2$

  $\Rightarrow x^2-mx-c=0$

  $\Rightarrow x=\frac{m\pm \sqrt{m^2+4c}}{2}$

If $m^2+4c>0,$ roots are real and distinct, the curves intersect at two points.

If $m^2+4c=0,$ roots are real and equal, the curves intersect at only one point.

If $m^2+4c<0,$ roots are imaginary, the curves never intersect.

Hence, the intersection of parabola and line depends on the nature of roots of the polynomial of degree 2.