CUET UG 2025 — General Test Geometry

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

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

$$P = \left(\frac{m \times x_2 + n \times x_1}{m+n}, \frac{m \times y_2 + n \times y_1}{m+n}\right)$$P=(m+nm×x2​+n×x1​​,m+nm×y2​+n×y1​​)

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Given:
$$(x_1, y_1) = (4, -3)$$(x1​,y1​)=(4,−3)
$$(x_2, y_2) = (8, 5)$$(x2​,y2​)=(8,5)
$m:n = 3:1$m:n=3:1, so $m = 3$m=3 and $n = 1$n=1

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For the x-coordinate:

$$x = \frac{m \times x_2 + n \times x_1}{m+n}$$x=m+nm×x2​+n×x1​​

$$x = \frac{3 \times 8 + 1 \times 4}{3+1}$$x=3+13×8+1×4​

$$x = \frac{24 + 4}{4}$$x=424+4​

$$x = \frac{28}{4}$$x=428​

$$x = 7$$x=7

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For the y-coordinate:

$$y = \frac{m \times y_2 + n \times y_1}{m+n}$$y=m+nm×y2​+n×y1​​

$$y = \frac{3 \times 5 + 1 \times (-3)}{3+1}$$y=3+13×5+1×(−3)​

$$y = \frac{15 - 3}{4}$$y=415−3​

$$y = \frac{12}{4}$$y=412​

$$y = 3$$y=3

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