The equation for f(x) is missing. We assume the equation of f(x) as:
y = (3/4)x.
The equation of the line that is perpendicular to f(x) where f(x) is given by y = (3/4)x is 4x + 3y = -2. Hence, option a is the right choice.
What is the equation of a line?
The equation of a line is the relation between x and y which gives us all the points (x, y) on the given line.
The equation of a line can be represented in two forms:
- Standard form: ax + by = c, where a, b, and c are constants.
- Slope-Intercept form: y = mx + b, where m is the slope of the line and b is the y-intercept.
How do we solve the given question?
In the question, we are asked to find the equation of the line that is perpendicular to f(x) where f(x) is y = (3/4)x and passes through the point (4, -6).
Given f(x): y = (3/4)x, is in the slope-intercept form. Hence, the slope of f(x), m₁ = (3/4).
Since the required line and f(x) are perpendicular to each other, the products of their slopes give the value -1.
Assuming the slope of the required line to be m, we can say that
m*m₁ = -1
or, m*(3/4) = -1
or, m = (-1)/(3/4) = -(4/3).
The slope of the required line is (-4/3).
Now, we know that a line having a slope = m, and passing through the point (x₁, y₁) can be represented by the equation,
y - y₁ = m(x - x₁).
Since the required line has a slope = (-4/3) and passes through the point (4, -6), we can write its equation as:
y - (-6) = (-4/3)(x - 4)
or, y + 6 = -(4/3)x + (4/3)(4)
or, (4/3)x + y = 16/3 - 6
or, (4x + 3y)/3 = (16 - 18)/3
or, 4x + 3y = -2.
∴ The equation of the line that is perpendicular to f(x) where f(x) is given by y = (3/4)x is 4x + 3y = -2. Hence, option a is the right choice.
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