Question

Find the coordinates of the point C on AB such that the ratio of AC to CB is 1:4 for A(-3,-1) and B(7,4)

Answer 1

Answer: C(-1,0)

Explanation:

A(-3,-1) B(7,4) AC:CB=1:4

`If\ you\ know\ two\ points\ of\ the\ plane\ A(x_A,y_A),\ B(x_B,y_B) \ then\ the\`

`coordinates \ of \ the\ point\ C(x_C,y_C),\ which\ divides\ the \ segment \ in\ the\ relation`

`\lambda=(AC)/(CB)\ are\ expressed \ by\ the\ formulas:`

`\boxed {x_C=(x_A+\lambda x_B)/(1+\lambda) }\ \ \ \ \ \ \ \ \ \ \ \ \boxed {y_C=(y_A+\lambda y_B)/(1+\lambda) }`

Hence,

`\displaystyle\\\lambda=(AC)/(CB)=(1)/(4) =0.25`

`\displaystyle\\x_C=(-3+0.25*7)/(1+0.25 )\\\\x_C=(-3+1.75)/(1.25) \\\\x_C=(-1,25)/(1.25) \\\\x_C=-1`

`\displaystyle\\y_C=(-1+0.25*4)/(1+0.25) \\\\y_C=(-1+1)/(1.25)\\\\y_C=(0)/(1.25)\\\\y_C=0`

Thus, C(-1,0)