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Answer :
Final answer:
The equation of L2 is y = (1/2)x - 9/2. The length of AB is sqrt(18).
Explanation:
To find the equation of line L2, we need to determine the slope of L2. Since L2 is perpendicular to L1, the slopes of the two lines are negative reciprocals of each other. The equation of line L1 is given as 2x + y = 8. We can rearrange this equation into the slope-intercept form, y = -2x + 8. The slope of L1 is -2.
Since L2 is perpendicular to L1, its slope will be the negative reciprocal of -2, which is 1/2. We can now use the point-slope form of a line to find the equation of L2. Using the point A (7, -4), the equation of L2 is y + 4 = (1/2)(x - 7), or y = (1/2)x - 9/2.
To find the length of AB, we can use the distance formula. The coordinates of A are (7, -4), and we know that B is the point of intersection of L1 and L2. We can solve the system of equations formed by the two lines and find the coordinates of B. Substituting the equation of L2 into the equation of L1, we get 2x + (1/2)x - 9/2 = 8. Solving for x, we find x = 4. Substituting this value back into the equation of L2, we find y = -1/2. Therefore, the coordinates of B are (4, -1). Using the distance formula, we can find the length of AB, which is sqrt[(4 - 7)^2 + (-1 - (-4))^2]. Simplifying, the length of AB is sqrt(18).
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