Question

The ratio in which the line 2y – 3x = 8 divides the line segment joining the points (1, 4) and (– 2, 3) is:-

Don't dare for Spam✨​

Answer 1

$\begin{gathered}\huge\blue{\mid{\fbox{\tt{SOLUTION}}\mid}} \\ \end{gathered}$

༒︎♡︎ EXPLANATION ♡︎༒︎

Ratio in which the line 2y - 3x = 8 divides the line segment joining the points (1, 4) and (-2,3).

As we know that,

Section formula for internal division.

⟹m+nmx2+nx1 , m+nmy2+ny1[/tex]

Using this formula in the equation, we get.

Let the line 2y - 3x = 8 divides the line segment in the ratio = k : 1.

⇒ x₁ = 1   and   y₁ = 4.

⇒ x₂ = - 2   and   y₂ = 3.

⇒ m = k   and   n = 1.

Put the values in the equation, we get.

$\sf \implies \dfrac{k(-2) + 1(1)}{k + 1} \ , \ \dfrac{k(3) + 1(4)}{k + 1} \\ ⟹k+1k(−2)+1(1) , k+1k(3)+1(4)$

$\sf \implies \dfrac{-2k + 1}{k + 1} \ , \ \dfrac{3k + 4}{k + 1} \\ ⟹k+1−2k+1 , k+13k+4$

Equation of line : 2y - 3x = 8.

Put the values of x and y in the equation of line, we get.

$\sf \implies 2 \bigg[ \dfrac{3k + 4}{k + 1} \bigg] - 3 \bigg[ \dfrac{-2k + 1}{k + 1} \bigg] \\ = 8⟹2[k+13k+4]−3[k+1−2k+1] \\ =8$

$\sf \implies \bigg[ \dfrac{6k + 8}{k + 1} \bigg] - \bigg[ \dfrac{- 6k + 3}{k + 1} \bigg] \\ = 8⟹[k+16k+8]−[k+1−6k+3] \\ =8$

$\sf \implies (6k + 8) - (-6k + 3) = 8(k + 1) \\ ⟹(6k+8)−(−6k+3)=8(k+1)$

$\sf \implies 6k + 8 + 6k - 3 = 8k + 8 \\ ⟹6k+8+6k−3=8k+8$

$\sf \implies 12k + 5 = 8k + 8 \\ ⟹12k+5=8k+8$

$\sf \implies 12k - 8k = 8 - 5 \\ ⟹12k−8k=8−5$

$\sf \implies 4k = 3 \\ ⟹4k=3$

$\sf \implies k = \dfrac{3}{4} \\ ⟹k=43$

Ratio = 3 : 4.

Answer 2

Answer:

3:4

Step-by-step explanation:

Given that the ratio in which the line 2y - 3x = 8 divides the line segment joining the points (1, 4) and (-2, 3) in ratio k:1.

Used formula: (mx2 + nx1)/(m + n), (my2 ny1)/(m + n)

Where, x 1 = 1, x2 = -2, y1 = 4, y2 = 3, m = k, n = 1

Substitute the values,

→ (-2k + 1)/(k + 1), (3k + 4)/(k + 1)

Equation: 2y - 3x = 8

→ 2 × (3k + 4)/(k + 1) - 3 × (-2k + 1)/(k + 1) = 8

→ (6k + 8)/(k + 1) + (6k - 3)/(k + 1) = 8

→ 6k + 8 + 6k - 3 = 8(k + 1)

→ 12k + 5 = 8k + 8

→ 12k - 8k = 8 - 5

→ 4k = 3

→ k = 3/4

Hence, the ratio is 3:4.