Question

1-63. the diagram at right shows a flat surface containing a line and a circle with no points in common. can you visualize moving the line and/or circle so that they intersect at exactly one point? two points? three points? explain each answer and illustrate each with an example when possible.

Explanation:

Step1: Analyze Intersection at One Point

A line and a circle can intersect at one point when the line is tangent to the circle. For example, if we have a circle with center \( O \) and radius \( r \), and a line \( l \) such that the distance from \( O \) to \( l \) is equal to \( r \), then the line is tangent to the circle and they intersect at exactly one point. Mathematically, the distance \( d \) from the center \((h,k)\) of the circle \((x - h)^2+(y - k)^2=r^2\) to the line \( Ax+By + C = 0\) is given by \( d=\frac{\vert Ah + Bk + C\vert}{\sqrt{A^2 + B^2}}\). When \( d = r \), the line is tangent (one intersection point).

Step2: Analyze Intersection at Two Points

A line and a circle can intersect at two points when the line is a secant of the circle. This happens when the distance from the center of the circle to the line is less than the radius (\( d

$$\begin{cases}(x - 0)^2+(y - 0)^2=25\\y = 0\end{cases}$$

\), substituting \( y = 0 \) into the circle equation gives \( x^2=25\), so \( x=\pm5 \), and the intersection points are \((5,0)\) and \((- 5,0)\).

Step3: Analyze Intersection at Three Points

A line is a one - dimensional object and a circle is a two - dimensional closed curve. A line can intersect a circle at at most two points. This is because the equation of a circle \((x - h)^2+(y - k)^2=r^2\) is a quadratic equation in \( x \) (or \( y \)) and the equation of a line \( y=mx + c \) (or \( x = a \)) is linear. Substituting the linear equation into the circle's equation gives a quadratic equation \( ax^2+bx + c = 0\) (in general form), and a quadratic equation has at most two real roots. So, a line and a circle cannot intersect at three points.

Answer:

• Exactly one point: Yes. When the line is tangent to the circle (distance from circle's center to line equals radius). Example: A circle \( x^{2}+y^{2}=4 \) (center \((0,0)\), radius \( 2\)) and the line \( y = 2 \) (distance from \((0,0)\) to \( y = 2 \) is \( 2 \), equal to radius), they intersect at \((0,2)\).
• Exactly two points: Yes. When the line is a secant (distance from circle's center to line is less than radius). Example: Circle \( x^{2}+y^{2}=9 \) (center \((0,0)\), radius \( 3\)) and line \( y = 0 \), intersect at \((3,0)\) and \((-3,0)\).
• Exactly three points: No. A line is linear (degree 1) and a circle is quadratic (degree 2). Substituting line equation into circle equation gives a quadratic equation, which has at most 2 real roots, so a line and a circle can intersect at most 2 times.