Help interpreting question about a system of equationsFor fixed a , b , m, and n, solve the system of equations: x 2 a 2 + y 2 b 2 = 1 and m x + n y = kchoosing k so that the resulting quadratic equation has a double root. I know how to solve a system of equations, but I'm not entirely sure what they're asking for here. I'm not really sure how I'd even end up with a factorable quadratic. Maybe just add the two together and try to factor?

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Help interpreting question about a system of equationsFor fixed   a  ,  b  ,  m, and   n, solve the system of equations:            x      2              a      2        +            y      2              b      2        =  1 and   m  x  +  n  y  =  kchoosing   k so that the resulting quadratic equation has a double root. I know how to solve a system of equations, but I&#039;m not entirely sure what they&#039;re asking for here. I&#039;m not really sure how I&#039;d even end up with a factorable quadratic. Maybe just add the two together and try to factor?

Kathryn Moody · Accepted Answer

I would say that by the resulting quadratic the meaning is this: when you solve (e.g) y from the lower equation and plug it in the first you get a quadratic in one variable.Now these equations represent an ellipse and a line. It&#039;s nice to think it geometrically too.            x      2              a      2        +            y      2              b      2        =  1 and   m  x  +  n  y  =  k  m  x  +  n  y  =  k  (      1          a      2        +            m      2                      b        2                    n        2              )      x    2    −            2      k      m                      b        2                    n        2              x  +            k      2                      b        2                    n        2              −  1  =  0For this to have double root, the discriminant must be zero:            4              k        2                    m        2                            b        4                    n        4              −  4  (      1          a      2        +            m      2                      b        2                    n        2              )  (            k      2                      b        2                    n        2              −  1  )  =  0  (            4              m        2                            b        4                    n        4              −            4      (              1                  a          2                    +                        m          2                                      b            2                                n            2                              )                      b        2                    n        2              )      k    2    +  4  (      1          a      2        +            m      2                      b        2                    n        2              )  =  0Solving this for   k gives  |  k  |  =                    4        (                  1                      a            2                          +                              m            2                                              b              2                                      n              2                                      )                                          4            (                          1                              a                2                                      +                                          m                2                                                              b                  2                                                  n                  2                                                      )                                              b              2                                      n              2                                      −                              4                          m              2                                                          b              4                                      n              4                                            =                    (                  1                      a            2                          +                              m            2                                              b              2                                      n              2                                      )                                          1                          a              2                                                          b              2                                      n              2                                      +                              m            2                                              b              4                                      n              4                                      −                              m            2                                              b              4                                      n              4                                            =            (              1                  a          2                    +                        m          2                                      b            2                                n            2                              )              1              a        b        n              =            a      2              m      2        +          b      2              n      2

Help interpreting question about a system of equationsFor fixed a , b , m , and

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