Let [dy dx = ax - by + a bx + cy + a], where a, b, c are constants, represent a circle pas…

Question

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 :

Accepted Answer

$${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$$

$$ = bx\,dy + cy\,dy + a\,dy = ax\,dx - by\,dx + a\,dx$$

$$ = cy\,dy + a\,dy - ax\,dx - a\,dx + b(x\,dy + y\,dx) = 0$$

$$ = c\int {y\,dy + a\int {x\,dx - a\int {dx + b\int {d(xy) = 0} } } } $$

$$ = {{c{y^2}} \over 2} + ay - {{a{x^2}} \over 2} - ax + bxy = k$$

$$ = a{x^2} - c{y^2} + 2ax - 2ay - 2bxy = k$$

Above equation is circle

$$\Rightarrow$$ a = - c and b = 0

$a{x^2} + a{y^2} + 2ax - 2ay = k$

$$ \Rightarrow {x^2} + {y^2} + 2x - 2y = \lambda \,\,\,\,\,\,\,\left[ {\lambda  = {k \over a}} \right]$$

Passes through (2, 5)

$$4 + 25 + 4 - 10 = \lambda  \Rightarrow \lambda  = 23$$

Circle $$ \equiv {x^2} + {y^2} + 2x - 2y - 23 = 0$$

Centre (-1, 1) $$r = \sqrt {{{( - 1)}^2} + {1^2} + 23}  = 5$$

Shortest distance of $$(11,6) = \sqrt {{{12}^2} + {5^2}}  - 5$$

$= 13 - 5$

$= 8$

Solutions · 1