Let dydx=ax-by+abx+cy+a, where a,b,c are constants, represent a circle through (2,5).

Question

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

Accepted Answer

Cross-multiplying and integrating: $ax^2-cy^2+2ax-2ay-2bxy=k$. For this to represent a circle, $a=-c$, $b=0$, giving
$$x^2+y^2+2x-2y=\lambda.$$

Through $(2,5)$: $4+25+4-10=\lambda \Rightarrow \lambda=23$. Circle: center $(-1,1)$, $r=\sqrt{1+1+23}=5$.

Distance from $(11,6)$ to center: $\sqrt{144+25}=13$. Shortest distance $=13-5=8$.

Solutions · 1