CUET UG 2025 — Mathematics Algebra

For three points to be collinear, they must lie on the same straight line. This means the slope between any two pairs of points must be equal.

Given points: $(a, b)$(a,b), $(c, d)$(c,d), and $(a + c, b + d)$(a+c,b+d)

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Slope from $(a, b)$(a,b) to $(a + c, b + d)$(a+c,b+d):

$$\text{Slope}_1 = \frac{(b+d) - b}{(a+c) - a}$$Slope1​=(a+c)−a(b+d)−b​

$$\text{Slope}_1 = \frac{d}{c}$$Slope1​=cd​

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Slope from $(c, d)$(c,d) to $(a + c, b + d)$(a+c,b+d):

$$\text{Slope}_2 = \frac{(b+d) - d}{(a+c) - c}$$Slope2​=(a+c)−c(b+d)−d​

$$\text{Slope}_2 = \frac{b}{a}$$Slope2​=ab​

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Since the points are collinear, the slopes must be equal:

$$\frac{d}{c} = \frac{b}{a}$$cd​=ab​

Cross-multiplying:

$$ad = bc$$ad=bc

Therefore, $ad = bc$ad=bc.