Answer:
To find the equation of a line perpendicular to the given line and passing through the point (3, 5), we need to determine the slope of the given line and then find the negative reciprocal of that slope, as perpendicular lines have slopes that are negative reciprocals of each other.
Given line: y = (-1/6)x^9
The slope of this line is -1/6.
The negative reciprocal of -1/6 is 6.
Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point (3, 5) and m is the slope 6, we can write the equation of the perpendicular line:
y - 5 = 6(x - 3)
Expanding and simplifying:
y - 5 = 6x - 18
y = 6x - 13
Therefore, the equation of the line perpendicular to the given line and passing through the point (3, 5) is y = 6x - 13.
To find the equation of the line parallel to the given line and passing through the point (3, 5), we can directly use the slope of the given line, which is -1/6.
Using the point-slope form, we can write the equation:
y - 5 = (-1/6)(x - 3)
Expanding and simplifying:
y - 5 = (-1/6)x + 1/2
y = (-1/6)x + 11/2
Therefore, the equation of the line parallel to the given line and passing through the point (3, 5) is y = (-1/6)x + 11/2.
Explanation:
To find the equation of a line that is perpendicular to the given line and passes through the point (3, 5), we need to determine the slope of the given line and then find the negative reciprocal of that slope, as perpendicular lines have slopes that are negative reciprocals of each other.
Given line: y = (-1/6)x^9
The slope of this line can be determined by comparing it to the standard slope-intercept form y = mx + b, where m represents the slope. In this case, the slope is (-1/6)^9.
The negative reciprocal of (-1/6)^9 will be the slope of the perpendicular line.
So, the slope of the perpendicular line is the negative reciprocal of (-1/6)^9, which can be calculated as (-6)^9.
Using the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the point (3, 5) and m is the slope we calculated (-6)^9, we can write the equation of the perpendicular line:
y - 5 = (-6)^9(x - 3)
Expanding and simplifying:
y - 5 = (-6)^9x - (-6)^9 * 3
y - 5 = (-6)^9x + C1 (where C1 is a constant)
Finally, we can rewrite the equation in the standard form:
(-6)^9x - y + C1 - 5 = 0
(-6)^9x - y + C2 = 0 (where C2 = C1 - 5)
Therefore, the equation of the line perpendicular to the given line and passing through the point (3, 5) is (-6)^9x - y + C2 = 0.
Now, let's find the equation of the line that is parallel to the given line and passes through the point (3, 5).
Since the parallel line will have the same slope as the given line, the slope will be (-1/6)^9.
Using the point-slope form, we can write the equation:
y - 5 = (-1/6)^9(x - 3)
Expanding and simplifying:
y - 5 = (-1/6)^9x - (-1/6)^9 * 3
y - 5 = (-1/6)^9x + C3 (where C3 is a constant)
Rewriting in the standard form:
(-1/6)^9x - y + C3 - 5 = 0
(-1/6)^9x - y + C4 = 0 (where C4 = C3 - 5)
Therefore, the equation of the line parallel to the given line and passing through the point (3, 5) is (-1/6)^9x - y + C4 = 0.
Note: The value of (-1/6)^9 is a very small number, and the resulting equation may not be easily interpretable.