Question

Let C be a circle of radius $\sqrt{20}$ cm. Let L1, L2 be the lines given by $2x - y - 1 = 0$ and $x + 2y - 18 = 0$, respectively. Suppose that L1 passes through the center of C and that L2 is tangent to C at the point of intersection of L1 and L2. If $(a,b)$ is the center of C, which of the following is a possible value of $a + b$?

Accepted Answer

B. Intersection P of L1 ($2x - y = 1$) and L2 ($x + 2y = 18$) is $P(4, 7)$. Center $(a,b)$ lies on L1, so $b = 2a - 1$. Distance from center $(a,b)$ to $P(4,7)$ equals radius $\sqrt{20}$: $(4-a)^2 + (7-b)^2 = 20 \Rightarrow (4-a)^2 + (8-2a)^2 = 20 \Rightarrow 5(4-a)^2 = 20 \Rightarrow (4-a)^2 = 4$. So $a = 6$ or $a = 2$. If $a = 6$, $b = 11 \Rightarrow a+b = 17$. If $a = 2$, $b = 3 \Rightarrow a+b = 5$. Possible value is 17 (Option B).

Circle Center and Tangent Line

Explanation

Practise this under exam conditions