Question

The slope of the tangent at the point (h,h) of the circle x2+y2=a2

Answer 1

The slope of tangent at point (h,h ) of the circle is -1

Answer 2

Slope of tangent at the point (h,h) to the circle $x^{2} + y^{2} = a^{2}$ is -1

• Given circle is

               $x^{2} + y^{2} = a^{2}$

• Equation of tangent at point $(x_{1}, y_{1})$ to the circle

         S ≡ $x^{2} + y^{2} + 2gx + 2fy + c = 0$ is given by

         $S_{1}$≡ $xx_{1} + yy_{1} + g(x + x_{1}) + f(y + y_{1}) + c = 0$

• From the given question, we have

          $(x_{1}, y_{1}) = (h, h)$, $g = 0, f = 0, c = a^{2}$

• Substituting above values in $S_{1} = 0$, we get tangent of equation as

          $x(h) + y(h) + 0(x + h) + 0(y + h) - a^{2} = 0\\ hx + hy - a^{2} = 0$

• Comparing equation of tangent with equation of line ax + by + c = 0

        a = h, b = h, c = $-a^{2}$

• Slope of line is given by

            m = -a/b = -h/h

         ∴ m = -1