If P(1,-2) is a point onthe linesegment A( x,y) and B (-2,4) such asth

1.	If P(1,-2) is a point onthe linesegment A( x,y) and B (-2,4) such asthat AP:PB= 2:3 find the coordinatesof A
Answer» We're given, $\longrightarrow \sf{A(x, y) = ({x}_{1}, {y}_{1})}$ $\longrightarrow \sf{P(1, -2) = (\alpha, \beta)}$ $\longrightarrow \sf{B(-2, 4) = ({x}_{2}, {y}_{2})}$ $\longrightarrow \sf{AP:PB = m:n = 2:3}$ $\sf{\\ \\}$ We KNOW by section formula, when a POINT $\sf{P(\alpha, \beta)}$ INTERSECTS internally a line segment AB where $\sf{A({x}_{1},{y}_{1})}$ and $\sf{B({x}_{2},{y}_{2})}$ in the RATIO m : n, then:- $\quad \quad \longrightarrow \sf{\alpha = \dfrac{m{x}_{2} + n{x}_{1}}{m + n}}$ $\quad \quad \longrightarrow \sf{\beta = \dfrac{m{y}_{2} + n{y}_{1}}{m + n}}$ $\sf{\\ \\}$ So, putting the VALUES, For alpha, $\longrightarrow \sf{\alpha = \dfrac{m{x}_{2} + n{x}_{1}}{m + n}}$ $\longrightarrow \sf{1 = \dfrac{(2 * -2) + (3 * x)}{2 + 3}}$ $\longrightarrow \sf{1 = \dfrac{ - 4 + 3x}{5}}$ $\longrightarrow \sf{5 = - 4 + 3x}$ $\longrightarrow \sf{9 = 3x}$ $\longrightarrow \underline{\sf{x = 3}}$ $\sf{\\ \\}$ For beta, $\longrightarrow \sf{\beta = \dfrac{m{y}_{2} + n{y}_{1}}{m + n}}$ $\longrightarrow \sf{-2 = \dfrac{(2 * 4) + (3 * y)}{2 + 3}}$ $\longrightarrow \sf{-2 = \dfrac{ 8 + 3y}{5}}$ $\longrightarrow \sf{-10 = 8 + 3y}$ $\longrightarrow \sf{-18 = 3y}$ $\longrightarrow \underline{\sf{y = -6}}$ $\sf{\\ \\}$ Hence the answer is, $\longrightarrow{\underline{\underline{\sf{A = (3,-6)}}}}$

If P(1,-2) is a point onthe linesegment A( x,y) and B (-2,4) such asthat AP:PB= 2:3 find the coordinatesof A

Answer»

We're given,

$\longrightarrow \sf{A(x, y) = ({x}_{1}, {y}_{1})}$

$\longrightarrow \sf{P(1, -2) = (\alpha, \beta)}$

$\longrightarrow \sf{B(-2, 4) = ({x}_{2}, {y}_{2})}$

$\longrightarrow \sf{AP:PB = m:n = 2:3}$

$\sf{\\ \\}$

We KNOW by section formula, when a POINT $\sf{P(\alpha, \beta)}$ INTERSECTS internally a line segment AB where $\sf{A({x}_{1},{y}_{1})}$ and $\sf{B({x}_{2},{y}_{2})}$ in the RATIO m : n, then:-