If the variable line 3x+4y=α 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…

Question

If the variable line 3x+4y=α 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 α is

MARKS Content Team · Accepted Answer

Correct answer: 165. Given line 3x + 4y = α Given circles (x-1)^2+(y-1)^2=1 & (x-9)^2+(y-1)^2=4 According to the given information centres of both the circles must lie on opposite side of line. L_11 . L_22 < 0 ⇒ 3 +4 -α 27 + 4 - α <0 α ∈ 7, 31 . Line is neither touching nor intersecting that means perpendicular distance from centre to line must more than or equal to radius. ⇒ (3+4-α)/5 ≥ 1 & (27+4-α)/5 ≥ 2 ⇒ α ∈ (-∞, 2] ∪ [12,∞) & α ∈ (-∞, 21] ∪ [41,∞) ⇒ α ∈ 12, 21 Sum of integer values of α = 12 + 13+ --21=165.

If the variable line $3 x + 4 y = α$ 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 $α$ is

Solution

Relevant Questions

If the variable line 3x+4y=α 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 α is

Solution

Relevant Questions

If the variable line $3 x + 4 y = α$ 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 $α$ is