Question

Find the ratio in which the line x = -2 divides the line
segment joining (-6, -1) and (1,6). Find the coordinates
of the point of intersection.
(3)​

Answer 1

Answer:

Let us assume that x = -2 cuts the line segment in the ratio k : 1.

According to section formula,

$(x,y) = \dfrac{k(x_2) + 1(x_1)}{k+1} , \dfrac{k(y_2) + 1(y_1)}{k+1}$

According to the question,

• x₁ = -6
• x₂ = 1
• y₁ = -1
• y₂ = 6

Substituting the values in the formula we get,

$(x,y) = \dfrac{ k(1) + 1(-6)}{k+1}, \dfrac{k(-1) + 1(6)}{k+1}$

But since x = -2 lines cuts the line segment, the x-coordinate of the unknown point is -2.

Therefore substituting the x coordinate as -2, we get:

$\rightarrow \dfrac{k-6}{k+1} = -2\\\\\rightarrow k - 6 = -2 ( k + 1 )\\\\\rightarrow k - 6 = -2k - 2\\\\\rightarrow k + 2k = 6 - 2 \\\\\rightarrow 3k = 4\\\\\rightarrow k = \dfrac{4}{3}$

Substituting the value of k in y coordinate we get:

$\rightarrow \dfrac{ 6 - k }{ k + 1 } = y\\\\\\\rightarrow \dfrac{ 6 - \dfrac{4}{3}}{\dfrac{4}{3} + 1} = y\\\\\\\rightarrow \dfrac{\dfrac{14}{3}}{\dfrac{7}{3}} = \dfrac{14}{7} = y\\\\\\\rightarrow \boxed{y=2}$

Therefore the point of intersection is ( -2,2 )

Answer 2

$\huge{\underline{\underline{\red{\mathfrak{Answer :}}}}}$

let the line segment cuts the line in the ratio of p : 1

Now use Section Formula :

$\Large{\sf{(x \: , \: y) \: = \: \dfrac{p(x_2) \: + \: 1(x_1)}{p \: + \: 1} \: , \dfrac{p(y_2) \: + \: 1(y_1)}{p \: + \: 1}}}$

Where,

• x1 = -6
• x2 = 1
• y1 = -1
• y2 = 6

Put these values in equation

${\sf{(x,y) = \dfrac{ p(1) + 1(-6)}{p+1}, \dfrac{p(-1) + 1(6)}{p+1}}}$

As x = -2 Put this value in X-Coordinate Therefore substituting the x coordinate as -2,

Put these values in equation

${\sf{(x,y) = \dfrac{ p(1) + 1(-6)}{p+1}, \dfrac{p(-1) + 1(6)}{p+1}}}$

As x = -2 Put this value in X-Coordinate Therefore substituting the x coordinate as -2, we get :

$\rightarrow {\sf{-2 \: = \: \dfrac{p \: - \: 6}{p \: + \: 1}}}$$\rightarrow {\sf{-2(p + 1) \: = \: p \: - \: 6}}$

$\rightarrow {\sf{-2p \: -2 \: = \: p \: - \: 6}}$

$\rightarrow {\sf{p \: + \: 2p \: = \: 6 \: - \: 2}}$

$\rightarrow {\sf{3p \: = \: 4}}$

$\rightarrow {\sf{p \: = \: \dfrac{4}{3}}}$

Put Value of p in Y - Coordinate We get

:

$\Large \rightarrow {\sf{y \: = \: \dfrac{6 - p}{p + 1}}}$

$\rightarrow {\sf{y \: = \: \dfrac{6 - 1.33}{1 + 1.33}}}$

$\rightarrow {\sf{y \: = \: \dfrac{4.66}{2.33}}}$

$\rightarrow {\sf{y \: = \: 2}}$

$\LARGE{\underline{\boxed{\sf{y \: = \: 2}}}}$

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