Find an equation of the normal line to the parabola y=x^2−8x+1 that is parallel to the line x−6y =7.
To find the equation of the normal line to the parabola that is parallel to a given line, we need to find the slope of the normal line.
First, let's find the slope of the given line. Rewrite the equation x - 6y = 7 in slope-intercept form (y = mx + b).
x - 6y = 7
-6y = -x + 7
y = (1/6)x - 7/6
The slope of this line is 1/6.
Now, to find the slope of the normal line to the parabola, we need to find the derivative of the parabola equation y = x^2 - 8x + 1.
Differentiate y = x^2 - 8x + 1 with respect to x:
dy/dx = 2x - 8
The derivative gives us the slope of the tangent line to the parabola at any given point. To find the slope of the normal line, we take the negative reciprocal of the derivative at the point of tangency.
Let's find the x-coordinate of the point of tangency between the normal line and the parabola.
Set 2x - 8 = 1/6 and solve for x:
2x - 8 = 1/6
2x = 1/6 + 8
2x = 49/6
x = 49/12
Plug this x-value back into the original parabolic equation to find the y-coordinate:
y = (49/12)^2 - 8(49/12) + 1
Now we have the point of tangency as (49/12, y).
The derivative at x = 49/12 is:
dy/dx = 2(49/12) - 8
Now, find the negative reciprocal of this derivative to get the slope of the normal line:
slope of normal line = -1 / (2(49/12) - 8)
Simplify the expression:
slope of normal line = -1 / ((98/12) - 8)
slope of normal line = -1 / ((98/12) - (96/12))
slope of normal line = -1 / (2/12)
slope of normal line = -12/2
slope of normal line = -6
Now we have the slope of the normal line, and we know that it is parallel to the line x - 6y = 7.
Using the point-slope form of a line equation, we can write the equation of the normal line:
y - y₁ = m(x - x₁)
Using the point of tangency (49/12, y) and the slope of the normal line (-6), we can substitute these values into the equation:
y - y₁ = -6(x - x₁)
Substituting the values:
y - y = -6(x - 49/12)
0 = -6(x - 49/12)
Simplify the equation:
0 = -6x + 49
Finally, rearrange the equation to the standard form:
6x = 49
Therefore, the equation of the normal line to the parabola y = x^2 - 8x + 1 that is parallel to the line x - 6y = 7 is 6x = 49.