To solve the problem step by step, we will follow the instructions given in the video transcript and derive the necessary calculations.
### Step 1: Find Points A and B
The line equation given is:
\[ x + 2y = 1 \]
**Finding Point A (Intersection with x-axis):**
To find the intersection with the x-axis, set \( y = 0 \):
\[
x + 2(0) = 1 \implies x = 1
\]
Thus, the coordinates of point A are:
\[ A(1, 0) \]
**Finding Point B (Intersection with y-axis):**
To find the intersection with the y-axis, set \( x = 0 \):
\[
0 + 2y = 1 \implies 2y = 1 \implies y = \frac{1}{2}
\]
Thus, the coordinates of point B are:
\[ B(0, \frac{1}{2}) \]
### Step 2: Circle Passing through A, B, and Origin
The circle passes through points A, B, and the origin O(0, 0). The general equation of a circle can be expressed as:
\[
x^2 + y^2 + 2gx + 2fy = 0
\]
Since the circle passes through the origin (0,0), we can simplify it to:
\[
x^2 + y^2 + 2gx + 2fy = 0
\]
Substituting point A(1, 0):
\[
1^2 + 0^2 + 2g(1) + 2f(0) = 0 \implies 1 + 2g = 0 \implies g = -\frac{1}{2}
\]
Substituting point B(0, \(\frac{1}{2}\)):
\[
0^2 + \left(\frac{1}{2}\right)^2 + 2g(0) + 2f\left(\frac{1}{2}\right) = 0 \implies \frac{1}{4} + f = 0 \implies f = -\frac{1}{4}
\]
Thus, the equation of the circle is:
\[
x^2 + y^2 - x - \frac{1}{2}y = 0
\]
### Step 3: Find the Tangent at the Origin
The tangent to the circle at the origin can be found using the formula:
\[
T = 0 \implies x_1x + y_1y + g = 0
\]
where \( (x_1, y_1) = (0, 0) \) and \( g = -\frac{1}{2} \):
\[
0 \cdot x + 0 \cdot y - \frac{1}{2} = 0 \implies -\frac{1}{2} = 0
\]
This implies that the tangent line at the origin is:
\[
2x + y = 0
\]
### Step 4: Calculate Perpendicular Distances from A and B to the Tangent Line
**Distance from A(1, 0):**
Using the distance formula from a point \( (h, k) \) to the line \( Ax + By + C = 0 \):
\[
\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]
For point A(1, 0):
- \( A = 2, B = 1, C = 0 \)
\[
\text{Distance from A} = \frac{|2(1) + 1(0) + 0|}{\sqrt{2^2 + 1^2}} = \frac{|2|}{\sqrt{5}} = \frac{2}{\sqrt{5}}
\]
**Distance from B(0, \(\frac{1}{2}\)):**
For point B(0, \(\frac{1}{2}\)):
\[
\text{Distance from B} = \frac{|2(0) + 1\left(\frac{1}{2}\right) + 0|}{\sqrt{2^2 + 1^2}} = \frac{\left|\frac{1}{2}\right|}{\sqrt{5}} = \frac{1/2}{\sqrt{5}} = \frac{1}{2\sqrt{5}}
\]
### Step 5: Sum of Distances
Now, we sum the distances from A and B:
\[
\text{Total Distance} = \frac{2}{\sqrt{5}} + \frac{1}{2\sqrt{5}} = \frac{4}{2\sqrt{5}} + \frac{1}{2\sqrt{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2}
\]
### Final Answer
Thus, the sum of the lengths of the perpendiculars from points A and B to the tangent at the origin is:
\[
\frac{\sqrt{5}}{2}
\]
