If the variable line 3x+4y= lies between the two circles (x-1)^2+(y-1)^2=1 and (x-9)^2+(y-…

Question

If the variable line $3x+4y=\alpha$ lies between the two circles $(x-1)^2+(y-1)^2=1$ and $(x-9)^2+(y-1)^2=4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is \_\_\_\_\_\_\_\_.

Answer: 165

Accepted Answer

Distance from $(1,1)$: $\dfrac{|7-\alpha|}5$; tangency at $\alpha=2,12$. Distance from $(9,1)$: $\dfrac{|31-\alpha|}5$; tangency at $\alpha=21,41$.

The line lies strictly between the circles (not crossing either) for $\alpha\in[12,21]$.

$$\sum_{\alpha=12}^{21}\alpha=\frac{(12+21)\cdot10}2=165.$$

Solutions · 1