find the minimum distance from the point (4,1) to the parabola y^2=4x.
The distance z from (4,1) to (x,y) is
z = √((x-4)^2 + (y-1)^2)
= √((y^2/4 - 4)^2 + (y-1)^2)
= (1/4)√(y^4-16y^2-32y+272)
dz/dy = (1/2)(y^3-8y-8)/√(y^4-16y^2-32y+272)
dz/dy=0 when y^3-8y-8=0
y = -2 or y = 1±√5
The minimum z is at y=1+√5
x = (3+√5)/2
z = (25-5√5)/2
or:
let the point of minimum distance from A(4,1) be P(a,b)
slope AP = (b-1)/(a-4)
2y dy/dx = 4
dy/dx = 2/y
so at A the tangent has a slope of 2/b
but AP is a normal to the tangent at A , so the slopes of the tangent and the slope of AP must be negative reciprocals of each other, i.e.
(a-4)/(1-b) = 2/b
a-4 = 2(1-b)/b , but b^2 = 4a ---> a = b^2/4
b^2/4 - 4 = (2 - 2b)/b
b^3 - 16b = 8 - 8b
b^3 - 8b - 8 = 0
b = -2, or b = 1 ± √5
reaching the same stage as Steve's method
To find the minimum distance from a point to a curve, we need to determine the point on the curve that is closest to the given point.
In this case, the point is (4,1), and the curve is the parabola y^2=4x.
We can start by assuming that the point on the curve closest to (4,1) is (a, b). We want to find the values of a and b that satisfy this assumption.
Since the point (a, b) lies on the parabola, we can substitute these values into the equation of the parabola:
b^2 = 4a
Now, we can solve this equation for a in terms of b:
a = b^2/4
Next, we can find the equation of the line passing through (4,1) and (a, b). The equation of the line can be written as:
(y - 1) = ((b - 1) / (a - 4)) * (x - 4)
Simplifying this equation, we get:
y - 1 = ((b - 1) / (b^2/4 - 4)) * (x - 4)
y - 1 = ((b - 1) / (b^2/4 - 16/4)) * (x - 4)
y - 1 = ((b - 1) / (b^2 - 16)) * 4 * (x - 4)
To find the point on the curve closest to (4,1), the slope of the line formed by the tangent line to the parabola at (a, b) should be equal to the slope of the line connecting (4,1) and (a, b).
The slope of the tangent line can be found by differentiating the equation of the parabola with respect to x:
dy/dx = 2b * db/da = 2b * (1/2) * b^(-1/2) = b^(-1/2) * b = 1/sqrt(b)
The slope of the line connecting (4,1) and (a, b) can be found using the formula:
m = (b - 1) / (a - 4)
Setting the slopes equal to each other, we get:
1/sqrt(b) = (b - 1) / (a - 4)
Cross-multiplying and simplifying, we have:
(a - 4) * sqrt(b) = (b - 1)
Squaring both sides, we get:
(a - 4)^2 * b = (b - 1)^2
Substituting the value of a in terms of b, we have:
(b^2/4 - 4)^2 * b = (b - 1)^2
Expanding and simplifying, we get a quartic equation:
b^5 - 10b^3 + 25b - 16 = 0
Now, we can solve this quartic equation to find the values of b that satisfy it. We can use numerical methods, such as the Newton-Raphson method or root-finding algorithms in computational software, to solve this equation and find the values of b.
Once we have the values of b, we can substitute them back into the equation a = b^2/4 to find the corresponding values of a.
Finally, we can calculate the minimum distance between the point (4,1) and the parabola using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) are the coordinates of the point (4,1) and (x2, y2) are the coordinates of the point on the parabola.