Question

a point p(1,0) divides line segment ab in a ratio of 3:5 from endpoint a(4,3). determine the coordinates of point b.

Explanation:

Step1: Use section - formula for x - coordinate

Let \(A(x_1,y_1)=(4,3)\), \(P(x,y)=(1,0)\) and the ratio \(m:n = 3:5\). The section - formula for the x - coordinate of a point \(P\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Substitute \(x = 1\), \(x_1=4\), \(m = 3\) and \(n = 5\) into the formula: \(1=\frac{3x_2+5\times4}{3 + 5}\).
\[1=\frac{3x_2 + 20}{8}\]
\[8=3x_2+20\]
\[3x_2=8 - 20=-12\]
\[x_2=-4\]

Step2: Use section - formula for y - coordinate

The section - formula for the y - coordinate of a point \(P\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Substitute \(y = 0\), \(y_1 = 3\), \(m = 3\) and \(n = 5\) into the formula: \(0=\frac{3y_2+5\times3}{3 + 5}\).
\[0=\frac{3y_2+15}{8}\]
\[0=3y_2+15\]
\[3y_2=-15\]
\[y_2=-5\]

Answer:

\((-4,-5)\)