If each of the lines 5x+8y=13 and 4x-y=3 contains a diameter of the circle x^2+y^2-2(a^2-7…

Question

If each of the lines $5x+8y=13$ and $4x-y=3$ contains a diameter of the circle $x^2+y^2-2(a^2-7a+11)x-2(a^2-6a+6)y+b^3+1=0$, then:

Accepted Answer

The intersection of the two given lines is $(1,1)$, which must be the centre of the circle:
$$a^2-7a+11=1\Rightarrow a=2,5;\qquad a^2-6a+6=1\Rightarrow a=1,5.$$
Common value: $a=5$.

Circle: $x^2+y^2-2x-2y+b^3+1=0\Rightarrow(x-1)^2+(y-1)^2=1-b^3.$ For a real circle, $1-b^3>0\Rightarrow b^3<1\Rightarrow b\in(-\infty,1).$

Solutions · 1