Two circles each of radius 5 units touch each other at the point (1,2).

Question

Two circles each of radius $5$ units touch each other at the point $(1,2)$. If the equation of their common tangent is $4x+3y=10$ and $C_1(\alpha,\beta)$ and $C_2(\gamma,\delta)$, $C_1
e C_2$, are their centres, then $|(\alpha+\beta)(\gamma+\delta)|$ is equal to \_\_\_\_\_\_\_\_.

Answer: 40

Accepted Answer

Centres lie along the normal to $4x+3y=10$ through $(1,2)$, at distance $5$: unit normal $\left(\dfrac45,\dfrac35\right)$.

$$C_1=(1,2)+5\left(\frac45,\frac35\right)=(5,5),\qquad C_2=(1,2)-5\left(\frac45,\frac35\right)=(-3,-1).$$

$$|(\alpha+\beta)(\gamma+\delta)|=|(10)(-4)|=40.$$

Solutions · 1