Find a point on the parabola y = (x-4)^2, where the tangent is parallel to the chord joining (4,0) and (5,1). Solve this question using Lagrange's theorem.
Answer is (9/2,1/4)
To find the point on the parabola where the tangent is parallel to the chord joining (4,0) and (5,1), we can use Lagrange's theorem.
Step 1: Find the equation of the tangent line to the parabola at an arbitrary point (x, y) on the parabola.
The slope of the tangent line is equal to the derivative of the parabola's equation.
So, taking the derivative of y = (x-4)^2 with respect to x, we get dy/dx = 2(x-4).
Step 2: Set up the equation for the slope of the chord joining (4,0) and (5,1).
The slope of the chord can be found using the formula: (y2 - y1) / (x2 - x1).
Plugging in the given coordinates (4,0) and (5,1), we get: (1 - 0) / (5 - 4) = 1/1 = 1.
Step 3: Apply Lagrange's theorem.
According to Lagrange's theorem, if the tangent line to a curve is parallel to a given line, then the slopes of the two lines are equal.
So, setting the slope of the tangent line equal to the slope of the chord, we have:
2(x-4) = 1.
Step 4: Solve for x.
2x - 8 = 1,
2x = 1 + 8,
2x = 9,
x = 9/2.
Step 5: Find the corresponding y-coordinate.
To find the y-coordinate, substitute the value of x into the equation of the parabola:
y = (9/2 - 4)^2 = (-7/2)^2 = 49/4.
Therefore, the point on the parabola where the tangent line is parallel to the chord joining (4,0) and (5,1) is (9/2, 49/4).