Pair Of Straight Lines

A homogeneous equation ax^2 + 2hxy + by^2 = 0 always factors into two lines through the origin, and the angle between them follows from tan(theta) = 2 sqrt(h^2 - ab) / |a + b|. Here a = 6, b = 3, and 2h = 11 so h = 11/2. The sum of coefficients is a + b = 9, which fixes the denominator of the tangent expression. The numerator is 2 sqrt(h^2 - ab) = 2 sqrt((11/2)^2 - 6*3) = 2 sqrt(121/4 - 18) = 2 sqrt(49/4) = 2 * (7/2) = 7. Dividing by a + b = 9 gives tan(theta) = 7/9, so the acute angle between the pair of lines is tan^{-1}(7/9). Option tan^{-1}(5/9) misreads the discriminant h^2 - ab. Option tan^{-1}(5/3) ignores the denominator a + b. Option tan^{-1}(1) wrongly assumes perpendicular lines, which would require a + b = 0. The result tan^{-1}(7/9) is the acute angle since the ratio 7/9 is positive and less than 1, matching the discriminant-over-trace structure of the angle formula.