On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$
1. **Identify the intersection point \( P \) of the parabola \( K \) and the line \( y = x \):**

   The equation of the parabola is \( y = \frac{x^2}{d} \). To find the intersection with the line \( y = x \), we set:
   \[
   x = \frac{x^2}{d}
   \]
   Solving for \( x \), we get:
   \[
   x^2 = dx \implies x(x - d) = 0
   \]
   Thus, \( x = 0 \) or \( x = d \). Since \( P \) is different from the origin, we have \( x = d \). Therefore, the coordinates of \( P \) are \( (d, d) \).

2. **Find the equation of the tangent line to the parabola at \( P \):**

   The slope of the tangent line to the parabola \( y = \frac{x^2}{d} \) at \( (d, d) \) is given by the derivative:
   \[
   \frac{dy}{dx} = \frac{2x}{d}
   \]
   At \( x = d \), the slope is:
   \[
   \frac{2d}{d} = 2
   \]
   Therefore, the equation of the tangent line at \( P \) is:
   \[
   y - d = 2(x - d) \implies y = 2x - d
   \]

3. **Determine the coordinates of \( Q \), where the circle touches the \( y \)-axis:**

   The circle touches the \( y \)-axis at \( Q \) and the tangent line at \( P \). The tangent line intersects the \( y \)-axis at:
   \[
   x = 0 \implies y = -d
   \]
   Let the center of the circle be \( (0, c) \). Since the circle touches the \( y \)-axis, its radius is \( c \). The distance from the center to \( P \) is also \( c \), so:
   \[
   \sqrt{d^2 + (d - c)^2} = c
   \]
   Squaring both sides, we get:
   \[
   d^2 + (d - c)^2 = c^2 \implies d^2 + d^2 - 2dc + c^2 = c^2 \implies 2d^2 - 2dc = 0 \implies d^2 = dc \implies c = d
   \]
   Therefore, the coordinates of \( Q \) are \( (0, d) \).

4. **Find the area \( S_1 \) between the line through \( P \) and \( Q \) and the parabola:**

   The equation of the line through \( P \) and \( Q \) is:
   \[
   y = \frac{d - d}{d - 0}x + d = d
   \]
   The area \( S_1 \) is the area between the line \( y = d \) and the parabola \( y = \frac{x^2}{d} \) from \( x = 0 \) to \( x = d \):
   \[
   S_1 = \int_0^d \left( d - \frac{x^2}{d} \right) dx
   \]
   Evaluating the integral, we get:
   \[
   S_1 = \int_0^d d \, dx - \int_0^d \frac{x^2}{d} \, dx = d \left[ x \right]_0^d - \frac{1}{d} \left[ \frac{x^3}{3} \right]_0^d = d^2 - \frac{d^3}{3d} = d^2 - \frac{d^2}{3} = \frac{2d^2}{3}
   \]

5. **Find the area \( S_2 \) between the line \( y = d \) and the parabola:**

   The area \( S_2 \) is the area between the line \( y = d \) and the parabola \( y = \frac{x^2}{d} \) from \( x = 0 \) to \( x = d \):
   \[
   S_2 = \int_0^d \left( d - \frac{x^2}{d} \right) dx
   \]
   This is the same integral as for \( S_1 \), so:
   \[
   S_2 = \frac{2d^2}{3}
   \]

6. **Calculate the ratio \( \frac{S_1}{S_2} \):**

   Since \( S_1 = S_2 \), we have:
   \[
   \frac{S_1}{S_2} = \frac{\frac{2d^2}{3}}{\frac{2d^2}{3}} = 1
   \]

The final answer is \( \boxed{1} \).