Write the equation 4x + 5y – 3 = 0 in normal form. Then, find the length of the normal and the angle it makes with the positive x–axis.
From
4x + 5y – 3 = 0
Divide all coefficients by sqrt(4^2+5^2) to get
4x/sqrt(41) +5y/sqrt(41) - 3/sqrt(41) = 0
where
3/sqrt(41) = length of normal
cos(θ)=4/sqrt(41), and
sin(θ)=5/sqrt(41)
Note: the normal form is given by:
x cos(θ) + y sin(θ) - p =0
where |p|=length of normal.
To write the equation 4x + 5y – 3 = 0 in normal form, we need to rearrange it such that the coefficients of x and y are divided by the magnitude of the equation's normal vector.
First, let's identify the coefficients of x and y in the given equation:
x coefficient: 4
y coefficient: 5
Now, let's find the magnitude of the normal vector by using the Pythagorean theorem:
Magnitude of the normal vector = √(x coefficient^2 + y coefficient^2)
= √(4^2 + 5^2)
= √(16 + 25)
= √41
Therefore, the magnitude of the normal vector is √41.
Next, divide the coefficients of x and y by the magnitude of the normal vector to obtain the normal form:
(4/√41)x + (5/√41)y – 3/√41 = 0
Now, let's find the length of the normal.
The length of the normal is the distance from the given line to the origin. We can calculate this distance using the formula:
Length of the normal = |constant term| / magnitude of the normal vector
= |(-3)/√41|
= 3/√41
Therefore, the length of the normal is 3/√41.
To find the angle the normal makes with the positive x-axis, we use the formula:
Angle = arctan(y coefficient / x coefficient)
Angle = arctan(5/4)
Using a calculator, we find that the angle is approximately 51.34 degrees.
So, the length of the normal is 3/√41 and the angle it makes with the positive x-axis is approximately 51.34 degrees.
To write the equation 4x + 5y - 3 = 0 in normal form, we need to rearrange the equation to have the coefficients of x and y multiplied with positive integers.
Starting with 4x + 5y - 3 = 0, we can move the constant term (-3) to the other side by adding 3 to both sides:
4x + 5y = 3
Next, we divide the entire equation by the greatest common divisor (GCD) of the coefficients of x and y to make the coefficients integers. In this case, the GCD of 4 and 5 is 1, so we proceed as follows:
(4x + 5y)/1 = 3/1
4x + 5y = 3
Now, the equation 4x + 5y = 3 is in normal form.
To find the length of the normal, we can use the formula:
Length = √(A^2 + B^2)
Where A and B are the coefficients of x and y, respectively.
In this case, A = 4 and B = 5. Substituting these values into the formula, we get:
Length = √(4^2 + 5^2)
= √(16 + 25)
= √41
Therefore, the length of the normal is √41.
To find the angle the normal makes with the positive x-axis, we can use the formula:
Angle = tan^(-1)(|B/A|)
Where B and A are the coefficients of y and x, respectively.
In this case, B = 5 and A = 4. Substituting these values into the formula, we get:
Angle = tan^(-1)(|5/4|)
≈ tan^(-1)(1.25)
≈ 51.34 degrees (rounded to two decimal places)
Therefore, the angle the normal makes with the positive x-axis is approximately 51.34 degrees.