Find the coordinates of the point which divides the line segment joining the points (4, -3) and (8,5) in the ratio 3:1 internally?

The point divides the line segment joining $(4, -3)$ and $(8, 5)$ in the ratio $3:1$ internally.

Using the section formula for internal division, when a point divides a line segment joining two points in the ratio $m:n$ internally:

$P = \left(\frac{m \times x_2 + n \times x_1}{m+n}, \frac{m \times y_2 + n \times y_1}{m+n}\right)$

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Given:

$(x_1, y_1) = (4, -3)$

$(x_2, y_2) = (8, 5)$

$m:n = 3:1$, so $m = 3$ and $n = 1$

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For the x-coordinate:

$x = \frac{m \times x_2 + n \times x_1}{m+n}$

$x = \frac{3 \times 8 + 1 \times 4}{3+1}$

$x = \frac{24 + 4}{4}$

$x = \frac{28}{4}$

$x = 7$

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For the y-coordinate:

$y = \frac{m \times y_2 + n \times y_1}{m+n}$

$y = \frac{3 \times 5 + 1 \times (-3)}{3+1}$

$y = \frac{15 - 3}{4}$

$y = \frac{12}{4}$

$y = 3$

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Therefore, the coordinates are $(7, 3)$.