An ellipse passing through the origin has its foci (3,4) and (6,8). Then length of its semi-minor axis is b, then the value of `b/sqrt2` is

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript while providing clear explanations at each step.

### Step 1: Identify the foci and their coordinates
The foci of the ellipse are given as \( F_1(3, 4) \) and \( F_2(6, 8) \).

### Step 2: Calculate the distances from the origin to the foci
We need to find the distances from the origin (0, 0) to each of the foci.

- Distance from the origin to \( F_1(3, 4) \):
\[ d_1 = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

- Distance from the origin to \( F_2(6, 8) \):
\[ d_2 = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 3: Use the property of the ellipse
According to the definition of an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equal to \( 2a \), where \( a \) is the semi-major axis.

Thus, we have:
\[ d_1 + d_2 = 5 + 10 = 15 = 2a \implies a = \frac{15}{2} \]

### Step 4: Calculate the distance between the foci
Next, we need to find the distance between the two foci \( F_1 \) and \( F_2 \):
\[ d_{F_1F_2} = \sqrt{(6 - 3)^2 + (8 - 4)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Relate the distance between foci to \( a \) and \( e \)
The distance between the foci is given by \( 2c \), where \( c = ae \) and \( e \) is the eccentricity of the ellipse. Thus:
\[ 2c = 5 \implies c = \frac{5}{2} \]

### Step 6: Relate \( a \), \( b \), and \( c \)
We know that:
\[ c^2 = a^2 - b^2 \]
Substituting the values of \( a \) and \( c \):
\[ \left(\frac{5}{2}\right)^2 = \left(\frac{15}{2}\right)^2 - b^2 \]
\[ \frac{25}{4} = \frac{225}{4} - b^2 \]
\[ b^2 = \frac{225}{4} - \frac{25}{4} = \frac{200}{4} = 50 \]

### Step 7: Find \( b \)
Taking the square root of both sides:
\[ b = \sqrt{50} = 5\sqrt{2} \]

### Step 8: Calculate \( \frac{b}{\sqrt{2}} \)
Finally, we need to find the value of \( \frac{b}{\sqrt{2}} \):
\[ \frac{b}{\sqrt{2}} = \frac{5\sqrt{2}}{\sqrt{2}} = 5 \]

### Final Answer
Thus, the value of \( \frac{b}{\sqrt{2}} \) is \( 5 \).