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?

As mentioned in the question,

Lines L1 and L2 intersect at point P as shown in the figure.

On solving for x and y from equations

x + 2y - 18 = 0

2x - y - 1 = 0

We get x = 4 and y =7.

Given, radius = $$\sqrt{20}$$

Using the equation of a circle, we have

$$(4-a)^{2}$$ + $$(7-b)^{2}$$ = 20....(1)

The center of the circle will lie on the line: 2x - y - 1 = 0

a,b will satisfy this equation.

So 2a-b-1=0

b=2a-1

From equation 1...

$$(4-a)^{2}$$ + $$(7-b)^{2}$$ = 20

$$(4-a)^{2}$$ + $$(8-2a)^{2}$$ = 20

5$$(4-a)^{2}$$ = 20

a=6 or a=2

b=11 or b=3

The sum of the coordinates possible=6+11  or 2+3 

i.e. 17 or 5

Option B is one of the solution.