{"id":5242,"date":"2024-10-14T14:05:06","date_gmt":"2024-10-14T14:05:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=5242"},"modified":"2025-08-15T16:37:01","modified_gmt":"2025-08-15T16:37:01","slug":"circles-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/circles-learn-it-4\/","title":{"raw":"Circles: Learn It 4","rendered":"Circles: Learn It 4"},"content":{"raw":"

Intersection of a Circle and a Line<\/h2>\r\nNow that we've got a solid understanding of circles and how to write their equations, let\u2019s explore what happens when a circle intersects with a line. This is a common situation in algebra and geometry, and it\u2019s fascinating to see how these two shapes interact. By analyzing the intersection points, we can solve various real-world problems, from finding the shortest path to designing curves in engineering.\r\n

When we talk about the intersection of a circle and a line, we're essentially looking for points that satisfy both the equation of the circle and the equation of the line simultaneously. This creates a system of equations:<\/p>\r\n\r\n

    \r\n \t
  1. The circle equation: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/li>\r\n \t
  2. The line equation: [latex]y = mx + b[\/latex]<\/li>\r\n<\/ol>\r\n

    Solving this system will give us the points where the line meets the circle, if any.<\/p>\r\n\r\n

    How To: Find the Intersection Points<\/strong>\r\n
      \r\n \t
    1. Substitute the equation of the line into the equation of the circle<\/li>\r\n \t
    2. Simplify and rearrange to get a quadratic equation in terms of [latex]x[\/latex] (or [latex]y[\/latex])<\/li>\r\n \t
    3. Solve the quadratic equation<\/li>\r\n \t
    4. Use the solutions to find the corresponding [latex]y[\/latex]-values (or [latex]x[\/latex]-values)<\/li>\r\n<\/ol>\r\n<\/section>\r\n

      The nature of the solutions to the quadratic equation tells us about the intersection:<\/p>\r\n\r\n

        \r\n \t
      • No real solutions: The line doesn't intersect the circle<\/li>\r\n \t
      • One real solution: The line is tangent to the circle<\/li>\r\n \t
      • Two real solutions: The line crosses through the circle<\/li>\r\n<\/ul>\r\n
        \r\n

        types of solutions for the points of intersection of a circle and a line<\/h3>\r\nThere are [latex]3[\/latex] possible\u00a0solution sets for a system of equations involving a circle and a line.\r\n\r\n\"\"\r\n
          \r\n \t
        • No solution<\/strong>.\r\n
            \r\n \t
          • The line does not intersect the circle.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
          • One solution<\/strong>.\r\n
              \r\n \t
            • The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
            • Two solutions<\/strong>.\r\n
                \r\n \t
              • The line crosses the circle and intersects it at two points.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section>
                Find the intersection of the given circle and the given line by substitution.\r\n
                [latex]\\begin{gathered}{x}^{2}+{y}^{2}=5 \\\\ y=3x - 5 \\end{gathered}[\/latex]<\/div>\r\n

                [reveal-answer q=\"766252\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"766252\"]<\/p>\r\nOne of the equations has already been solved for [latex]y[\/latex]. We will substitute [latex]y=3x - 5[\/latex] into the equation for the circle.\r\n

                [latex]\\begin{gathered}{x}^{2}+{\\left(3x - 5\\right)}^{2}=5\\\\ {x}^{2}+9{x}^{2}-30x+25=5\\\\ 10{x}^{2}-30x+20=0\\end{gathered}[\/latex]<\/div>\r\nNow, we factor and solve for [latex]x[\/latex].\r\n
                [latex]\\begin{gathered}10\\left({x}^{2}-3x+2\\right)=0 \\\\ 10\\left(x - 2\\right)\\left(x - 1\\right)=0 \\\\ x=2 \\hspace{5mm} x=1 \\end{gathered}[\/latex]<\/div>\r\nSubstitute the two [latex]x[\/latex]-values into the original linear equation to solve for [latex]y[\/latex].\r\n
                [latex]\\begin{align}y&=3\\left(2\\right)-5 \\\\ &=1 \\\\[3mm] y&=3\\left(1\\right)-5 \\\\ &=-2 \\end{align}[\/latex]<\/div>\r\n
                \r\n\r\nThe line intersects the circle at [latex]\\left(2,1\\right)[\/latex] and [latex]\\left(1,-2\\right)[\/latex], which can be verified by substituting these [latex]\\left(x,y\\right)[\/latex] values into both of the original equations.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"400\"]\"Line Circle on a coordinate plane with a line intersecting two points[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
                [ohm2_question hide_question_numbers=1]24881[\/ohm2_question]<\/section>
                [ohm2_question hide_question_numbers=1]24882[\/ohm2_question]<\/section>","rendered":"

                Intersection of a Circle and a Line<\/h2>\n

                Now that we’ve got a solid understanding of circles and how to write their equations, let\u2019s explore what happens when a circle intersects with a line. This is a common situation in algebra and geometry, and it\u2019s fascinating to see how these two shapes interact. By analyzing the intersection points, we can solve various real-world problems, from finding the shortest path to designing curves in engineering.<\/p>\n

                When we talk about the intersection of a circle and a line, we’re essentially looking for points that satisfy both the equation of the circle and the equation of the line simultaneously. This creates a system of equations:<\/p>\n

                  \n
                1. The circle equation: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/li>\n
                2. The line equation: [latex]y = mx + b[\/latex]<\/li>\n<\/ol>\n

                  Solving this system will give us the points where the line meets the circle, if any.<\/p>\n

                  How To: Find the Intersection Points<\/strong><\/p>\n
                    \n
                  1. Substitute the equation of the line into the equation of the circle<\/li>\n
                  2. Simplify and rearrange to get a quadratic equation in terms of [latex]x[\/latex] (or [latex]y[\/latex])<\/li>\n
                  3. Solve the quadratic equation<\/li>\n
                  4. Use the solutions to find the corresponding [latex]y[\/latex]-values (or [latex]x[\/latex]-values)<\/li>\n<\/ol>\n<\/section>\n

                    The nature of the solutions to the quadratic equation tells us about the intersection:<\/p>\n

                      \n
                    • No real solutions: The line doesn’t intersect the circle<\/li>\n
                    • One real solution: The line is tangent to the circle<\/li>\n
                    • Two real solutions: The line crosses through the circle<\/li>\n<\/ul>\n
                      \n

                      types of solutions for the points of intersection of a circle and a line<\/h3>\n

                      There are [latex]3[\/latex] possible\u00a0solution sets for a system of equations involving a circle and a line.<\/p>\n

                      \"\"<\/p>\n

                        \n
                      • No solution<\/strong>.\n
                          \n
                        • The line does not intersect the circle.<\/li>\n<\/ul>\n<\/li>\n
                        • One solution<\/strong>.\n
                            \n
                          • The line is tangent to the circle and intersects the circle at exactly one point.<\/li>\n<\/ul>\n<\/li>\n
                          • Two solutions<\/strong>.\n
                              \n
                            • The line crosses the circle and intersects it at two points.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n
                              Find the intersection of the given circle and the given line by substitution.<\/p>\n
                              [latex]\\begin{gathered}{x}^{2}+{y}^{2}=5 \\\\ y=3x - 5 \\end{gathered}[\/latex]<\/div>\n

                              \n