The question asks to calculate the ratio in which the line segment joining the points $P(1, 8)$ and $Q(4, 2)$ is internally divided by the point $O(3, 4)$.

We can solve this problem using the section formula. If a point $O(x_1, y_1)$ divides the line joining $P(x_2, y_2)$ and $Q(x_3, y_3)$ in the ratio $m:n$, then the coordinates of $O$ can be expressed as:

$x_1 = \frac{m \cdot x_3 + n \cdot x_2}{m + n}, \quad y_1 = \frac{m \cdot y_3 + n \cdot y_2}{m + n}$

Here, $P(1, 8)$, $Q(4, 2)$, and $O(3, 4)$. Let the ratio be $m:n$.

Step 1: Apply the section formula for the x-coordinate

The x-coordinate of $O$ is 3, and using the section formula:

$3 = \frac{m \cdot 4 + n \cdot 1}{m + n}$

Multiply both sides by $m + n$:

$3(m + n) = 4m + 1n$

$3m + 3n = 4m + n$

Simplify:

$m = 2n$

Step 2: Apply the section formula for the y-coordinate

Now use the y-coordinate of $O$ (which is 4):

$4 = \frac{m \cdot 2 + n \cdot 8}{m + n}$

Multiply both sides by $m + n$:

$4(m + n) = 2m + 8n$

$4m + 4n = 2m + 8n$

Simplify:

$2m = 4n$

$m = 2n$

Step 3: Determine the ratio

From both equations, we see that $m = 2n$. Hence, the ratio in which point $O(3, 4)$ divides the line segment joining $P(1, 8)$ and $Q(4, 2)$ is:

$m:n = 2:1$

The correct answer is $2:1$, not $1:3$. It seems the given answer in the image may be incorrect.

Would you like more details or have any further questions?

Here are 5 related questions you might find helpful:

1. How is the section formula applied to find the dividing ratio for 3D points?
2. What is the difference between internal and external division in coordinate geometry?
3. How do you calculate the distance between two points in a 2D plane?
4. What is the significance of the midpoint in dividing a line segment?
5. How can the section formula be used in real-life applications?

Tip: Always verify both coordinates (x and y) when using the section formula to ensure consistency in solving problems involving dividing points.