. A line L1 passes through points (2, 4) and (-1, -5). Find(a) the equation of L1 in the form of ax + by + c = 0(b) the equation of L2 perpendicular to L1 and passes through (-2, 4)(c) the point of intersection between L1 and L2
Question
. A line L1 passes through points (2, 4) and (-1, -5). Find(a) the equation of L1 in the form of ax + by + c = 0(b) the equation of L2 perpendicular to L1 and passes through (-2, 4)(c) the point of intersection between L1 and L2
Solution
To find the equation of line L1, we can use the formula for the equation of a line passing through two points.
Step 1: Find the slope of line L1 using the formula: slope (m) = (y2 - y1) / (x2 - x1) Let's use the points (2, 4) and (-1, -5) to calculate the slope: m = (-5 - 4) / (-1 - 2) = -9 / -3 = 3
Step 2: Use the slope-intercept form of a line (y = mx + b) to find the y-intercept (b). We can choose either of the given points to substitute into the equation. Let's use (2, 4): 4 = 3(2) + b 4 = 6 + b b = 4 - 6 b = -2
Step 3: Write the equation of line L1 in the form ax + by + c = 0. Since the slope (m) is 3 and the y-intercept (b) is -2, the equation of line L1 is: 3x - y - 2 = 0
To find the equation of line L2, which is perpendicular to L1 and passes through (-2, 4), we need to find the negative reciprocal of the slope of L1.
Step 4: Find the negative reciprocal of the slope of L1. The slope of L1 is 3, so the negative reciprocal is -1/3.
Step 5: Use the point-slope form of a line (y - y1) = m(x - x1) to find the equation of L2. Let's use the point (-2, 4) and the negative reciprocal slope (-1/3): y - 4 = (-1/3)(x - (-2)) y - 4 = (-1/3)(x + 2) y - 4 = (-1/3)x - 2/3
Step 6: Write the equation of line L2 in the form ax + by + c = 0. Rearranging the equation, we get: (1/3)x + y - (10/3) = 0
To find the point of intersection between L1 and L2, we can solve the system of equations formed by the two lines.
Step 7: Set the equations of L1 and L2 equal to each other and solve for x and y. 3x - y - 2 = (1/3)x + y - (10/3) 9x - 3y - 6 = x + 3y - 10 8x - 6y = 4
Simplifying the equation, we get: 4x - 3y = 2
Step 8: Solve the system of equations to find the point of intersection. We can use various methods to solve the system, such as substitution or elimination. Let's use the elimination method.
Multiply the equation 4x - 3y = 2 by 3 to eliminate the y term: 12x - 9y = 6
Now, subtract the equation 8x - 6y = 4 from 12x - 9y = 6: (12x - 9y) - (8x - 6y) = 6 - 4 12x - 9y - 8x + 6y = 2 4x - 3y = 2
We obtain the same equation as before, which means the system is dependent and has infinitely many solutions. Therefore, the point of intersection between L1 and L2 is not a single point but a line.
In summary: (a) The equation of line L1 is 3x - y - 2 = 0. (b) The equation of line L2 is (1/3)x + y - (10/3) = 0. (c) The point of intersection between L1 and L2 is a line, not a single point.
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