Let ${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$, where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is :

<p>${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$</p> <p>$= bx\,dy + cy\,dy + a\,dy = ax\,dx - by\,dx + a\,dx$</p> <p>$= cy\,dy + a\,dy - ax\,dx - a\,dx + b(x\,dy + y\,dx) = 0$</p> <p>$= c\int {y\,dy + a\int {x\,dx - a\int {dx + b\int {d(xy) = 0} } } }$</p> <p>$= {{c{y^2}} \over 2} + ay - {{a{x^2}} \over 2} - ax + bxy = k$</p> <p>$= a{x^2} - c{y^2} + 2ax - 2ay - 2bxy = k$</p> <p>Above equation is circle</p> <p>$\Rightarrow$ a = $-$ c and b = 0</p> <p>$a{x^2} + a{y^2} + 2ax - 2ay = k$</p> <p>$$ \Rightarrow {x^2} + {y^2} + 2x - 2y = \lambda \,\,\,\,\,\,\,\left[ {\lambda = {k \over a}} \right]$$</p> <p>Passes through (2, 5)</p> <p>$4 + 25 + 4 - 10 = \lambda \Rightarrow \lambda = 23$</p> <p>Circle $\equiv {x^2} + {y^2} + 2x - 2y - 23 = 0$</p> <p>Centre ($-$1, 1) $r = \sqrt {{{( - 1)}^2} + {1^2} + 23} = 5$</p> <p>Shortest distance of $(11,6) = \sqrt {{{12}^2} + {5^2}} - 5$</p> <p>$= 13 - 5$</p> <p>$= 8$</p>