Question

Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, -3). Hence find m.​

Answer 1

Answer:

here is your answer

P(4,3) ; A(2,-) ; B(6,-3)

let the ratio be = k : 1

P (4,m) = k (6) + 1(2) / k+1

k(-3 + 1(3) / k+1

according the problem,,

4 = 6k +2 / k+1

=>2(k +1) = 3k +1

=>k = 1

and also,,

m = -3(1) +3 / (1) +1

=>m = 0/2

=>m = 0

so, your question is completed.

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Answer 2

P(4,m) divides the line segment AB joining points A(2,3) and B(6,-3).

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☯ Let P(4,m) divides line segment AB in ratio k : 1.

Now, Using Section formula,

$\star\;{\boxed{\sf{\pink{(x,y) = \bigg( \dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2} , \dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \bigg)}}}}$

where,

• $\bf x_1 , y_1$ = Coordinates of first point

• $\bf x_2 , y_2$ = Coordinates of second point

• $\bf m_1 , m_2$ = Ratio at which line is divided

⠀

Putting values,

$:\implies\sf (4, m) = \bigg( \dfrac{k \times 6 + 1 \times 2}{k + 1} , \dfrac{k \times - 3 + 1 \times 3}{k + 1} \bigg)$

$\qquad \quad:\implies\sf (4, m) = \bigg( \dfrac{6k + 2}{k + 1} , \dfrac{-3k + 3}{k + 1} \bigg)$

$\qquad\quad:\implies\sf 4 = \dfrac{6k + 2}{k + 1} , m = \dfrac{-3k + 3}{k + 1}$

Therefore,

$\qquad\qquad:\implies\sf 4(k + 1) = 6k + 2$

$\qquad\qquad\: \: :\implies\sf 4k + 4 = 6k + 2$

$\qquad\qquad\: :\implies\sf 4 - 2 = 6k - 4k$

$\qquad\qquad\: \: \quad\quad:\implies\sf 2 = 2k$

$\qquad\qquad\: \: \qquad:\implies\sf k = \cancel{ \dfrac{2}{2}}$

$\qquad\qquad\qquad:\implies{\underline{\boxed{\frak{\purple{k = 1}}}}}\;\bigstar$

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$\therefore$ P divides the line segment AB in ratio 1:1.

Also, P is the mid - point of AB.

☯ Now, Putting value of k in,

$\qquad\qquad \qquad:\implies\sf m = \dfrac{-3k + 3}{k + 1}$

$\qquad\qquad\quad:\implies\sf m = \dfrac{-3 \times 1 + 3}{1 + 1}$

$\qquad\qquad\qquad:\implies\sf m = \dfrac{-3 + 3}{2}$

$\qquad\qquad\qquad\quad:\implies\sf m = \dfrac{0}{2}$

$\qquad\qquad\qquad\quad:\implies{\underline{\boxed{\frak{\purple{m = 0}}}}}\;\bigstar$

$\therefore$ Hence, The value of m is 0.

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

⠀⠀⠀⠀⠀☯ Distance Formula which is used to find the distance between two defined points.

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