Question

PLS SOLVE NO SPAMMING​

Answer 1

$\huge{\bf{\green{\mathfrak{Question:-}}}}$

$\bullet{\rightarrow}$ If the point C (- 1,2) divides internally the line segment joining A (2,5) and B(x, y) in the ratio 3:4 then, find the coordinates of B.

$\huge {\bf{\orange{\mathfrak{Answer:-}}}}$

$\bullet{\rightarrow} \: \textsf{Coordinates of Line segment = A(2,5) and B(x,y).}$

$\bullet{\rightarrow}$ $\textsf{C(- 1,2) divides AB in the ratio 3:4.}$

$\bullet{\rightarrow} \: \sf Ratio \: = \: m_{1}:m_{2} \: = \: 3:4.$

$\bullet{\rightarrow}$ $\textsf{We know that,}$

$\bullet{\rightarrow} \: \boxed{\sf Section \: Formula \: = \: \dfrac{m_{1} \: * \: x_{2} \: + \: m_{2} \: * \: x_{1}}{m_{1} \: + \: m_{2}} \: , \: \dfrac{m_{1} \: * \: y_{2} \: + \: m_{2} \: * \: y_{1}}{m_{1} \: + \: m_{2}}}$

$\bullet{\rightarrow} \: \textsf{Substituting the values of ratio and coordinates,}$

$\bullet{\rightarrow} \: \sf C(- \: 1,2) \: = \: \dfrac{3 \: * \: x \: + \: 4 \: * \: 2}{3 \: + \: 4} \: , \: \dfrac{3 \: * \: y \: + \: 4 \: * \: 5}{3 \: + \: 4}$

$\bullet{\rightarrow}$ $\sf C(- \: 1,2) \: = \: \dfrac{3x \: + \: 8}{7} \: , \: \dfrac{3y \: + \: 20}{7}$

$\bullet{\rightarrow}$ $\textsf{Equating the coordinates,}$

$\bullet{\rightarrow}$ $\sf - \: 1 \: = \: \dfrac{3x \: + \: 8}{7} \: , \: 2 \: = \: \dfrac{3y \: + \: 20}{7}$

$\bullet{\rightarrow}$ $\sf - \: 1 \: * \: 7 \: = \: 3x \: + \: 8 \: , \: 2 \: * \: 7 \: = \: 3y \: + \: 20$

$\bullet{\rightarrow}$ $\sf - \: 7 \: = \: 3x \: + \: 8 \: , \: 14 \: = \: 3y \: + \: 20$

$\bullet{\rightarrow}$ $\sf - \: 7 \: - \: 8 \: = \: 3x \: , \: 14 \: - \: 20 \: = \: 3y$

$\bullet{\rightarrow}$ $\sf - \: 15 \: = \: 3x \: , \: - \: 6 \: = \: 3y$

$\bullet{\rightarrow}$ $\sf x \: = \: \dfrac{- \: 15}{3} \: , \: y \: = \: \dfrac{- \: 6}{3}$

$\bullet{\rightarrow}$ $\boxed{\sf x \: = \: - \: 5 \: , \: y \: = \: - \: 2.}$

$\huge{\bf{\red{\mathfrak{Conclusion:-}}}}$

$\bullet{\rightarrow}\: \boxed{\therefore{\sf Coordinates \: of \: B \: are \: x \: = \: - \: 5 \: , \: y \: = \: - \: 2.}}$

$\bullet{\rightarrow}$ $\boxed{\sf B(- \: 5,\: - 2) \: is \: the \: answer.}$

Answer 2

Answer:

☯ Let the Co-ordinate of point B be (x, y) and AC: BC = 3:4.

$\star$ DIAGRAM:

⠀⠀$\setlength{\unitlength}{14mm}\begin{picture}(7,5)(0,0)\thicklines\put(0,0){\line(1,0){5}}\put(5.1, - 0.3){\sf B}\put( - 0.2, - 0.3){\sf A}\put(5.2, 0){\sf (x, y)}\put( - 0.7, 0){\sf (2,5)}\put(2.3, 0.2){\sf C}\put(2.2, - 0.3){\sf (-1,2)}\put(5, 0){\circle*{0.1}}\put(2.4, 0){\circle*{0.1}}\put(0, 0){\circle*{0.1}}\put(1,0.2){\sf 3}\put(3.5, 0.2){\sf 4}\end{picture}$⠀⠀

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

⠀⠀⠀

Given that,

The point C(-1, 2) divide internally the line segment joining points A(2, 5) and B in the ratio 3:4.

$\underline{\bigstar\:\boldsymbol{Using\:section\:formula\::}}$

$\dag\:\boxed{\sf{\pink{\Big(x, y \Big) = \Bigg(\dfrac{mx_2 + nx_1}{m + n} \dfrac{my_2 + ny_1}{m + n}\Bigg)}}}$

$:\implies\sf \dfrac{3x + 4(2)}{3 + 4} = -1 \\\\\\:\implies\sf \dfrac{3x + 8}{7} \\\\\\\\:\implies\sf 3x + 8 = -7 \\\\\\:\implies\sf x = \cancel\dfrac{-15}{\:3} \\\\\\:\implies{\underline{\boxed{\frak{\purple{x = -5}}}}}\:\bigstar$

Similarly,

⠀⠀⠀

$:\implies\sf \dfrac{3y + 4(5)}{3 + 4} = 2\\\\\\:\implies\sf \dfrac{3y + 20}{7} = 2 \\\\\\:\implies\sf 3y + 20 = 14\\\\\\:\implies\sf 3y = -6\\\\\\:\implies\sf y = \cancel\dfrac{-6}{\:3}\\\\\\:\implies{\underline{\boxed{\frak{\purple{y = -2}}}}}\:\bigstar$

⠀⠀

$\therefore\:{\underline{\sf{Hence, \ the \ Co-ordinate \ of \ point \ B \ is\: \bf{\Big(-5, -2 \Big)}.}}}$