Find the geometric place of the points from where as we draw the tangents at ellipse x 2 a 2 + y 2 b 2 = 1 , they are perpendicular.

Question

Find the geometric place of the points from where as we draw the tangents at ellipse             x      2              a      2        +            y      2              b      2        =  1 , they are perpendicular.

Amy Daniels · Accepted Answer

Step 1The tangent line with slope m, to the ellipse             x      2              a      2        +            y      2              b      2        =  1 is given as :  y  =  m  x  ±            a              2                    m              2              +          b              2            and the tangent line perpendicular to it, with slope   −      1    m   is given as :  y  =  −      1    m    x  ±                    a        2                    m        2              +          b              2              . .Now change each equation to make :  y  −  m  x  =  ±            a              2                    m              2              +          b              2            and  m  y  +  x  =  ±            a              2              +          m              2                    b              2              . .Now square each equation and add them :  m  y  +  x  =  ±            a              2              +          m              2                    b              2              .Therefore we get a circle where      x          2        +      y          2        =      a          2        +      b          2        . .This is called a director circle of an ellipse.

Find the geometric place of the points from where as we draw the tangents at ellipse

Answered question

Answer & Explanation

New Questions in Elementary geometry