Question

The line y = 2 - 2x cuts the curve 3x^2 - y^2 = 3 at the points A and B. Find the length of the line AB.

Option 1: 2√5
Option 2: 4√2
Option 3: 3√3
Option 4: 5√2

Answer 1

To find the length of the line segment AB, first equate the line equation with the curve equation and solve for x. The length can then be found by taking absolute difference of the two x values. The answer is Option 2: 4 sqrt(2).

Step-by-step explanation:

To solve this question, we first need to equate the line equation with the curve equation. Thus, replacing y = 2 - 2x in the curve equation, we get 3x^2 - (2 - 2x)^2 = 3, which simplifies to 5x^2 - 4x - 2 = 0. Solving this quadratic equation using the quadratic formula, we get x = [4 ± sqrt(16 + 40)] / 10 = [4 ± sqrt(56)] / 10 = [4 ± 2 sqrt(14)] / 10 = 2/5 ± sqrt(14) / 5. The length of the line segment AB can be found by |2 sqrt(14) / 5 - (-2 sqrt(14) / 5)| = 4 sqrt(14) / 5. Converting it into simplest form, the length is 4 sqrt(2). Thus, the correct answer is Option 2: 4 sqrt(2).

Learn more about Length of a Line Segment