Question

The ratio in which x – axis divides the line segment joining the points (5, 4) and (2, –3) is:

(A) 5 : 2 (B) 3 : 4 (C) 2 : 5 (D) 4 : 3​

Answer 1

☯ Let the points A(5,4) and B(2,-3) is divided by x - axis in the ratio m : n.

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

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Given that,

• The points lies on x - axis. So, y cordinate is 0.

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

$\star\;{\boxed{\sf{\pink{(x,y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\;,\; \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}}$

$\sf Here \begin{cases} & \sf{x_1 , y_1 = 5,4} \\ & \sf{x_2 , y_2 = 2,-3} \\ & \sf{x,y = x,0} \end{cases}$

Therefore,

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$:\implies\sf y = \dfrac{m \times (-3) + n \times (4)}{m + n}$

$:\implies\sf 0 = \dfrac{-3m + 4n}{m + n}$

$:\implies\sf - 3m + 4n = 0 \times (m + n)$

$:\implies\sf - 3m + 4n = 0$

$:\implies\sf -3m = - 4n$

$:\implies\sf 3m = 4n$

$:\implies\sf \dfrac{m}{n} = \dfrac{4}{3}$

$:\implies{\underline{\boxed{\sf{\purple{m : n = 4 : 3}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{The\;ratio\; in \;which \;x - axis\; divides\; the \;line\; segment\; is\; {\textsf{\textbf{4 : 3}}}.}}}$

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$\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\: More\:to\:know:}}}}}\mid}$

• Distance Formula = The distance formula is a formula that is used to find the distance between two points.

• $\sf d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Answer 2

Step-by-step explanation:

☯ Let the points A(5,4) and B(2,-3) is divided by x - axis in the ratio m : n.

⠀⠀

⠀⠀━━━━━━━━━━━━━━━━━━━━━━

Given that,

The points lies on x - axis. So, y cordinate is 0.

⠀⠀

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

$\star\;{\boxed{\sf{\pink{(x,y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\;,\; \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}}$

$\sf Here \begin{cases} & \sf{x_1 , y_1 = 5,4} \\ & \sf{x_2 , y_2 = 2,-3} \\ & \sf{x,y = x,0} \end{cases}$

Therefore,

⠀⠀

$:\implies\sf y = \dfrac{m \times (-3) + n \times (4)}{m + n}$

$:\implies\sf 0 = \dfrac{-3m + 4n}{m + n}$

$:\implies\sf - 3m + 4n = 0 \times (m + n)$

$:\implies\sf - 3m + 4n = 0$

$:\implies\sf -3m = - 4n$

$:\implies\sf 3m = 4n$

$:\implies\sf \dfrac{m}{n} = \dfrac{4}{3}$

$:\implies{\underline{\boxed{\sf{\purple{m : n = 4 : 3}}}}}\;\bigstar$

$\therefore\;{\underline{\sf{The\;ratio\; in \;which \;x - axis\; divides\; the \;line\; segment\; is\; {\textsf{\textbf{4 : 3}}}.}}}$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━

$\qquad\qquad\boxed{\bf{\mid{\overline{\underline{\pink{\bigstar\: More\:to\:know:}}}}}\mid}$

Distance Formula = The distance formula is a formula that is used to find the distance between two points.

$\sf d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$