find the equation to the straight line which passes through the point of intersection of the line 3x+4y-1=0 and 5x+8y-3=0 and is perpendicular to the line 4x-2y+3=0
the two lines intersect at (-1,1)
The required slope is -1/2
SO, the desired line is
y-1 = -1/2 (x+1)
To find the equation of a straight line passing through the point of intersection of two given lines and perpendicular to another given line, you need to follow these steps:
1. Find the point of intersection of the two given lines.
2. Determine the slope of the given line.
3. Find the negative reciprocal of the slope from step 2 to obtain the slope of the line you want to find.
4. Use the slope from step 3 and the point of intersection from step 1 to form the equation of the line using the point-slope form or the slope-intercept form.
Let's go through the steps to find the equation of the desired line:
Step 1: Finding the point of intersection:
The point of intersection is the solution to the system of equations formed by the two given lines. We can solve this system of equations using any method (substitution, elimination, or matrices). In this case, let's use the substitution method.
Equation 1: 3x + 4y - 1 = 0
Equation 2: 5x + 8y - 3 = 0
From Equation 1, solve for x:
3x = 1 - 4y
x = (1 - 4y) / 3
Substitute the value of x into Equation 2:
5((1 - 4y) / 3) + 8y - 3 = 0
Simplify the equation and solve for y:
5 - 20y/3 + 8y - 3 = 0
-20y/3 + 8y + 2 = 0
-20y + 24y + 6 = 0
4y = -6
y = -6/4
y = -3/2
Substitute the value of y into Equation 1 to find x:
3x + 4(-3/2) - 1 = 0
3x - 6 - 1 = 0
3x - 7 = 0
3x = 7
x = 7/3
Therefore, the point of intersection is (7/3, -3/2).
Step 2: Determine the slope of the given line:
The given line to which the desired line is perpendicular has an equation of the form ax + by + c = 0, where a, b, and c are constants. The slope of this line can be found using the formula: slope = -a/b.
From the equation 4x - 2y + 3 = 0, we can see coefficients a = 4 and b = -2.
Therefore, the slope of the given line is -a/b = -4/(-2) = 2.
Step 3: Find the negative reciprocal of the slope from step 2:
The negative reciprocal of 2 is -1/2.
Step 4: Write the equation of the desired line using the slope from step 3 and the point of intersection from step 1:
Using the point-slope form, the equation of the desired line is:
(y - y₁) = m(x - x₁)
where (x₁, y₁) is the point of intersection (7/3, -3/2) and m is the slope (-1/2).
Substituting the values, we have:
(y - (-3/2)) = (-1/2)(x - 7/3)
(y + 3/2) = (-1/2)(x - 7/3)
2(y + 3/2) = - (x - 7/3)
Multiplying both sides of the equation by 2 to eliminate fractions:
2y + 3 = -2(x - 7/3)
Simplifying the equation further:
2y + 3 = -2x + 14/3
2y = -2x + 14/3 - 9/3
2y = -2x + 5/3
Thus, the equation of the straight line passing through the point of intersection of the lines 3x + 4y - 1 = 0 and 5x + 8y - 3 = 0 and perpendicular to the line 4x - 2y + 3 = 0 is 2y = -2x + 5/3.