Equation of tangent on Cartesian plane given center and radius of a circle












0












$begingroup$


If I have a generic circle with radius $r$ and center $(h, k)$, and a tangent line with point of tangency $(x, y)$, can you give me the equation of the tangent line? Thanks!










share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    If I have a generic circle with radius $r$ and center $(h, k)$, and a tangent line with point of tangency $(x, y)$, can you give me the equation of the tangent line? Thanks!










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      If I have a generic circle with radius $r$ and center $(h, k)$, and a tangent line with point of tangency $(x, y)$, can you give me the equation of the tangent line? Thanks!










      share|cite|improve this question











      $endgroup$




      If I have a generic circle with radius $r$ and center $(h, k)$, and a tangent line with point of tangency $(x, y)$, can you give me the equation of the tangent line? Thanks!







      geometry trigonometry circle coordinate-systems






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited May 31 '14 at 0:17









      Cookie

      8,720123682




      8,720123682










      asked May 31 '14 at 0:15









      CarraraCarrara

      323




      323






















          2 Answers
          2






          active

          oldest

          votes


















          0












          $begingroup$

          Here's how I was taught to get tangents to conic sections in high school, half a century ago. If you want to get the tangent at the point $(x_0,y_0)$, you take the equation of the conic, and replace "half" of the $x$'s by $x_0$ and "half" of the $y$'s by $y_0$.



          In this case, the equation of the curve is $(x-h)^2+(y-k)^2=r^2$, so the tangent at $(x_0,y_0)$ will be $(x_0-h)(x-h) + (y_0-k)(y-k)=r^2$.



          A little algebra, using the fact that $x_0^2 + y_0^2 = r^2$, shows that this can also be written in the form $(x_0 -h)(x-x_0) + (y_0 -k)(y-y_0) = 0$.



          This works with any conic section curve, and it gives you tangent planes of quadric surfaces, too.



          Of course, you can apply the same process even if the point $(x_0,y_0)$ doesn't lie on the curve. In this case, the line you get is called the polar line of the point. It just happens that, for any point lying on the curve, the polar line is the tangent at the point.



          I'm curious -- do they still teach this stuff?






          share|cite|improve this answer











          $endgroup$





















            0












            $begingroup$

            Let the point of tangency be $P=(x_0, y_0)$. If the center of the circle is at $C=(h,k)$, then the line $overleftrightarrow{CP}$ is perpendicular to tangent line at the point $P$. The slope of the line $overleftrightarrow{CP}$ is $dfrac{y_0-k}{x_0-h}$, so the slope of the tangent line is $m=-dfrac{x_0-h}{y_0-k}$.



            The equation of the tangent line is therefore



            $$y - y_0 = -dfrac{x_0-h}{y_0-k}(x-x_0)$$



            Just to satisfy my own curiosity, I want to write this equation is standard form.



            begin{align}
            y - y_0 &= -dfrac{x_0-h}{y_0-k}(x-x_0) \
            (y_0-k)y - (y_0-k)y_0 &= -(x_0-h)x + (x_0-h)x_0 \
            (x_0-h)x + (y_0-k)y &= (x_0-h)x_0 + (y_0-k)y_0 \
            (x_0-h)x + (y_0-k)y &= x_0^2 + y_0^2 -h x_0 - k y_0\
            end{align}






            share|cite|improve this answer











            $endgroup$













              Your Answer





              StackExchange.ifUsing("editor", function () {
              return StackExchange.using("mathjaxEditing", function () {
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              });
              });
              }, "mathjax-editing");

              StackExchange.ready(function() {
              var channelOptions = {
              tags: "".split(" "),
              id: "69"
              };
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function() {
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled) {
              StackExchange.using("snippets", function() {
              createEditor();
              });
              }
              else {
              createEditor();
              }
              });

              function createEditor() {
              StackExchange.prepareEditor({
              heartbeatType: 'answer',
              autoActivateHeartbeat: false,
              convertImagesToLinks: true,
              noModals: true,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              imageUploader: {
              brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
              contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
              allowUrls: true
              },
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              });


              }
              });














              draft saved

              draft discarded


















              StackExchange.ready(
              function () {
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f815506%2fequation-of-tangent-on-cartesian-plane-given-center-and-radius-of-a-circle%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              0












              $begingroup$

              Here's how I was taught to get tangents to conic sections in high school, half a century ago. If you want to get the tangent at the point $(x_0,y_0)$, you take the equation of the conic, and replace "half" of the $x$'s by $x_0$ and "half" of the $y$'s by $y_0$.



              In this case, the equation of the curve is $(x-h)^2+(y-k)^2=r^2$, so the tangent at $(x_0,y_0)$ will be $(x_0-h)(x-h) + (y_0-k)(y-k)=r^2$.



              A little algebra, using the fact that $x_0^2 + y_0^2 = r^2$, shows that this can also be written in the form $(x_0 -h)(x-x_0) + (y_0 -k)(y-y_0) = 0$.



              This works with any conic section curve, and it gives you tangent planes of quadric surfaces, too.



              Of course, you can apply the same process even if the point $(x_0,y_0)$ doesn't lie on the curve. In this case, the line you get is called the polar line of the point. It just happens that, for any point lying on the curve, the polar line is the tangent at the point.



              I'm curious -- do they still teach this stuff?






              share|cite|improve this answer











              $endgroup$


















                0












                $begingroup$

                Here's how I was taught to get tangents to conic sections in high school, half a century ago. If you want to get the tangent at the point $(x_0,y_0)$, you take the equation of the conic, and replace "half" of the $x$'s by $x_0$ and "half" of the $y$'s by $y_0$.



                In this case, the equation of the curve is $(x-h)^2+(y-k)^2=r^2$, so the tangent at $(x_0,y_0)$ will be $(x_0-h)(x-h) + (y_0-k)(y-k)=r^2$.



                A little algebra, using the fact that $x_0^2 + y_0^2 = r^2$, shows that this can also be written in the form $(x_0 -h)(x-x_0) + (y_0 -k)(y-y_0) = 0$.



                This works with any conic section curve, and it gives you tangent planes of quadric surfaces, too.



                Of course, you can apply the same process even if the point $(x_0,y_0)$ doesn't lie on the curve. In this case, the line you get is called the polar line of the point. It just happens that, for any point lying on the curve, the polar line is the tangent at the point.



                I'm curious -- do they still teach this stuff?






                share|cite|improve this answer











                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Here's how I was taught to get tangents to conic sections in high school, half a century ago. If you want to get the tangent at the point $(x_0,y_0)$, you take the equation of the conic, and replace "half" of the $x$'s by $x_0$ and "half" of the $y$'s by $y_0$.



                  In this case, the equation of the curve is $(x-h)^2+(y-k)^2=r^2$, so the tangent at $(x_0,y_0)$ will be $(x_0-h)(x-h) + (y_0-k)(y-k)=r^2$.



                  A little algebra, using the fact that $x_0^2 + y_0^2 = r^2$, shows that this can also be written in the form $(x_0 -h)(x-x_0) + (y_0 -k)(y-y_0) = 0$.



                  This works with any conic section curve, and it gives you tangent planes of quadric surfaces, too.



                  Of course, you can apply the same process even if the point $(x_0,y_0)$ doesn't lie on the curve. In this case, the line you get is called the polar line of the point. It just happens that, for any point lying on the curve, the polar line is the tangent at the point.



                  I'm curious -- do they still teach this stuff?






                  share|cite|improve this answer











                  $endgroup$



                  Here's how I was taught to get tangents to conic sections in high school, half a century ago. If you want to get the tangent at the point $(x_0,y_0)$, you take the equation of the conic, and replace "half" of the $x$'s by $x_0$ and "half" of the $y$'s by $y_0$.



                  In this case, the equation of the curve is $(x-h)^2+(y-k)^2=r^2$, so the tangent at $(x_0,y_0)$ will be $(x_0-h)(x-h) + (y_0-k)(y-k)=r^2$.



                  A little algebra, using the fact that $x_0^2 + y_0^2 = r^2$, shows that this can also be written in the form $(x_0 -h)(x-x_0) + (y_0 -k)(y-y_0) = 0$.



                  This works with any conic section curve, and it gives you tangent planes of quadric surfaces, too.



                  Of course, you can apply the same process even if the point $(x_0,y_0)$ doesn't lie on the curve. In this case, the line you get is called the polar line of the point. It just happens that, for any point lying on the curve, the polar line is the tangent at the point.



                  I'm curious -- do they still teach this stuff?







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited May 31 '14 at 6:54

























                  answered May 31 '14 at 4:10









                  bubbabubba

                  30.3k33086




                  30.3k33086























                      0












                      $begingroup$

                      Let the point of tangency be $P=(x_0, y_0)$. If the center of the circle is at $C=(h,k)$, then the line $overleftrightarrow{CP}$ is perpendicular to tangent line at the point $P$. The slope of the line $overleftrightarrow{CP}$ is $dfrac{y_0-k}{x_0-h}$, so the slope of the tangent line is $m=-dfrac{x_0-h}{y_0-k}$.



                      The equation of the tangent line is therefore



                      $$y - y_0 = -dfrac{x_0-h}{y_0-k}(x-x_0)$$



                      Just to satisfy my own curiosity, I want to write this equation is standard form.



                      begin{align}
                      y - y_0 &= -dfrac{x_0-h}{y_0-k}(x-x_0) \
                      (y_0-k)y - (y_0-k)y_0 &= -(x_0-h)x + (x_0-h)x_0 \
                      (x_0-h)x + (y_0-k)y &= (x_0-h)x_0 + (y_0-k)y_0 \
                      (x_0-h)x + (y_0-k)y &= x_0^2 + y_0^2 -h x_0 - k y_0\
                      end{align}






                      share|cite|improve this answer











                      $endgroup$


















                        0












                        $begingroup$

                        Let the point of tangency be $P=(x_0, y_0)$. If the center of the circle is at $C=(h,k)$, then the line $overleftrightarrow{CP}$ is perpendicular to tangent line at the point $P$. The slope of the line $overleftrightarrow{CP}$ is $dfrac{y_0-k}{x_0-h}$, so the slope of the tangent line is $m=-dfrac{x_0-h}{y_0-k}$.



                        The equation of the tangent line is therefore



                        $$y - y_0 = -dfrac{x_0-h}{y_0-k}(x-x_0)$$



                        Just to satisfy my own curiosity, I want to write this equation is standard form.



                        begin{align}
                        y - y_0 &= -dfrac{x_0-h}{y_0-k}(x-x_0) \
                        (y_0-k)y - (y_0-k)y_0 &= -(x_0-h)x + (x_0-h)x_0 \
                        (x_0-h)x + (y_0-k)y &= (x_0-h)x_0 + (y_0-k)y_0 \
                        (x_0-h)x + (y_0-k)y &= x_0^2 + y_0^2 -h x_0 - k y_0\
                        end{align}






                        share|cite|improve this answer











                        $endgroup$
















                          0












                          0








                          0





                          $begingroup$

                          Let the point of tangency be $P=(x_0, y_0)$. If the center of the circle is at $C=(h,k)$, then the line $overleftrightarrow{CP}$ is perpendicular to tangent line at the point $P$. The slope of the line $overleftrightarrow{CP}$ is $dfrac{y_0-k}{x_0-h}$, so the slope of the tangent line is $m=-dfrac{x_0-h}{y_0-k}$.



                          The equation of the tangent line is therefore



                          $$y - y_0 = -dfrac{x_0-h}{y_0-k}(x-x_0)$$



                          Just to satisfy my own curiosity, I want to write this equation is standard form.



                          begin{align}
                          y - y_0 &= -dfrac{x_0-h}{y_0-k}(x-x_0) \
                          (y_0-k)y - (y_0-k)y_0 &= -(x_0-h)x + (x_0-h)x_0 \
                          (x_0-h)x + (y_0-k)y &= (x_0-h)x_0 + (y_0-k)y_0 \
                          (x_0-h)x + (y_0-k)y &= x_0^2 + y_0^2 -h x_0 - k y_0\
                          end{align}






                          share|cite|improve this answer











                          $endgroup$



                          Let the point of tangency be $P=(x_0, y_0)$. If the center of the circle is at $C=(h,k)$, then the line $overleftrightarrow{CP}$ is perpendicular to tangent line at the point $P$. The slope of the line $overleftrightarrow{CP}$ is $dfrac{y_0-k}{x_0-h}$, so the slope of the tangent line is $m=-dfrac{x_0-h}{y_0-k}$.



                          The equation of the tangent line is therefore



                          $$y - y_0 = -dfrac{x_0-h}{y_0-k}(x-x_0)$$



                          Just to satisfy my own curiosity, I want to write this equation is standard form.



                          begin{align}
                          y - y_0 &= -dfrac{x_0-h}{y_0-k}(x-x_0) \
                          (y_0-k)y - (y_0-k)y_0 &= -(x_0-h)x + (x_0-h)x_0 \
                          (x_0-h)x + (y_0-k)y &= (x_0-h)x_0 + (y_0-k)y_0 \
                          (x_0-h)x + (y_0-k)y &= x_0^2 + y_0^2 -h x_0 - k y_0\
                          end{align}







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited May 16 '18 at 23:36

























                          answered May 16 '18 at 23:24









                          steven gregorysteven gregory

                          18k32258




                          18k32258






























                              draft saved

                              draft discarded




















































                              Thanks for contributing an answer to Mathematics Stack Exchange!


                              • Please be sure to answer the question. Provide details and share your research!

                              But avoid



                              • Asking for help, clarification, or responding to other answers.

                              • Making statements based on opinion; back them up with references or personal experience.


                              Use MathJax to format equations. MathJax reference.


                              To learn more, see our tips on writing great answers.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function () {
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f815506%2fequation-of-tangent-on-cartesian-plane-given-center-and-radius-of-a-circle%23new-answer', 'question_page');
                              }
                              );

                              Post as a guest















                              Required, but never shown





















































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown

































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown







                              Popular posts from this blog

                              Mario Kart Wii

                              What does “Dominus providebit” mean?

                              The Binding of Isaac: Rebirth/Afterbirth