In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7, 6)?

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$\frac{6k-3}{k+1}=0$

$6k-3=0$

$k=\frac{3}{6}=\frac{1}{2}$

Thus, x-axis divides the line segment joining the points (2, –3) and (5,6) in the ratio 1:2.

Let the point P(5, 4) divide the line segment joining the points (2, 1), (7, 6) in the ratio m1 : m2.

By Section-formula, we get the coordinates of point P as:

(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2).\Big(\dfrac{m1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big).(m1​+m2​m1​x2​+m2​x1​​,m1​+m2​m1​y2​+m2​y1​​).

Putting values in x coordinate of above equation we get,

=m1×7+m2×2m1+m2=7m1+2m2m1+m2= \dfrac{m1 \times 7 + m2 \times 2}{m1 + m2} \\[1em] = \dfrac{7m1 + 2m2}{m1 + m2} \\[1em]=m1​+m2​m1​×7+m2​×2​=m1​+m2​7m1​+2m2​​

According to question, the x-coordinate of P = 5. Comparing we get,

⇒7m1+2m2m1+m2=5⇒7m1+2m2=5m1+5m2⇒7m1−5m1=5m2−2m2⇒2m1=3m2⇒m1m2=32.\Rightarrow \dfrac{7m1 + 2m2}{m1 + m2} = 5 \\[1em] \Rightarrow 7m1 + 2m2 = 5m1 + 5m2 \\[1em] \Rightarrow 7m1 - 5m1 = 5m2 - 2m2 \\[1em] \Rightarrow 2m1 = 3m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{3}{2}.⇒m1​+m2​7m1​+2m2​​=5⇒7m1​+2m2​=5m1​+5m2​⇒7m1​−5m1​=5m2​−2m2​⇒2m1​=3m2​⇒m2​m1​​=23​.

Hence, the ratio in which point (5, 4) divides the line segment is 3 : 2.