Show that the numerical value of the radius of curvature at the point (x1, y1) on the parabola y^2=4ax is [2(a+x1)^3/2)]/a^1/2. If c is the centre of the curvature at the origin O and S is the point (a, 0), show that OC=2(OS)

plz show me working plz

r = (1+y'^2)^(3/2) / |y"|

y^2 = 4ax, so
y = 2√(ax)
y' = a/√(ax)
y" = -a^2/(2(ax)^(3/2))
= -a/(2x√(ax))

so just plug it in:

(1+y'^2) = 1+a/x
r = ((a+x)/x)^3/2 * 2x√(ax)/a
= 2(a+x)^(3/2) / √a

To find the numerical value of the radius of curvature at the point (x1, y1) on the parabola y^2 = 4ax, we can use the formula:

Radius of curvature (R) = [(1+(dy/dx)^2)^1.5] / |d^2y/dx^2|

First, let's find the first derivative of y = 2 √(ax):

dy/dx = 2√(a) * (1/2) * (x^-1/2) = √(a) / √(x)

Next, let's find the second derivative of y = 2 √(ax):

d^2y/dx^2 = (1/2) * √(a) * (-1/2) * x^(-3/2) = -√(a) / (2 √(x^3))

Now we have the derivatives, so we can substitute them back into the formula:

R = [(1 + (√(a) / √(x))^2)^1.5] / |-√(a) / (2 √(x^3))|

Simplifying the expression:

R = [1 + (a / x)]^1.5 / (√(a) / (2 √(x^3)))

R = 2√(x^3)(√(x^3) + a) / √(a)

Now, let's substitute x1 into the equation to find the radius of curvature at point (x1, y1):

R = 2√(x1^3)(√(x1^3) + a) / √(a)

R = 2(x1^3 / √(a))(√(x1^3) + a) / √(a)

R = 2(x1^3 + ax1) / a

Finally, we have the numerical value of the radius of curvature at the point (x1, y1) on the parabola y^2 = 4ax:

R = [2(a + x1)^3/2] / √(a)

To prove that OC = 2(OS), we need to find the coordinates of point C, which represents the center of curvature at the origin O.

The center of curvature is given by the equation (h, 0), where h is the x-coordinate of point C.

For a parabola y^2 = 4ax, the coordinates of point S are (a, 0). Since S and O are symmetric to the parabolic axis, the x-coordinate of point C is given by C = 2a - h.

We know that OC = 2(OS), so we can write it as:

|OC|^2 = 4|OS|^2

The magnitude squared of OC is given by:

|OC|^2 = (2a - h)^2

The magnitude squared of OS is given by:

|OS|^2 = (a - h)^2

Now, let's substitute these values into the equation:

(2a - h)^2 = 4(a - h)^2

Expanding both sides:

4a^2 - 4ah + h^2 = 4a^2 - 8ah + 4h^2

Simplifying the equation:

-4ah + h^2 = -4ah + 4h^2

Cancelling out terms:

h^2 = 0

Therefore, h = 0.

Since C = 2a - h, we have C = 2a - 0 = 2a.

Thus, OC = 2(OS), which proves the statement.