Question

.What ratio the line joining - 1,1 and 5,7 is divided by the line X + y is equal to 4​

Answer 1

Answer:

$1 : 2$

Step-by-step explanation:

The two given points are $A(-1,1), B (5,7)$

Using the two point form we can get the equation of the line joining $AB$

   $\frac{y - y_{1}}{y_{2} - y_{1}} = \frac{x-x_{1}}{x_{2} - x_{1}}$

⇒ $\frac{y - 1}{7-1} = \frac{x - (-1)}{5-(-1)}$

⇒ $\frac{y-1}{6} = \frac{x+1}{6}$

⇒ $y-1 = x+1$

⇒ $x-y = -2$                   $----(1)$

∴ The equation of the line joining $AB$ = $x-y = -2$

The equation of the dividing line = $x + y = 4$         $----(2)$

Now, we have to find the Point of Intersection $(POI)$ of the two equations

Adding $(1)$ and $(2)$ we get

   $x-y + x + y = -2 + 4$

⇒ $2x = 2$

⇒ $x = 1$

Putting the value of $x$ in either $(1)$ or $(2)$ we get

    $y = 3$

∴  $POI = (1,3)$

Let the ratio be $1:k$

Now we can apply the section formula

   $(\frac{mx_{2} + nx_{1}}{m+n} , \frac{my_{2} + ny_1}{m +n} )$

⇒ $(\frac{1(5) + k(-1)}{1+k},\frac{1(7) + k(1)}{1+k} ) = (1,3)$

⇒ $(\frac{5-k}{1+k} , \frac{7 + k}{1 + k} ) = (1,3)$

⇒  $\frac{5 - k}{1+k} = 1$ ; $\frac{7 + k}{1+k} = 3$

⇒ $5-k = 1(k+1)$ ; $7 + k = 3(k+1)$

⇒ $5 - k = k +1$  ; $7 + k = 3k + 3$

⇒ $5-k-k-1 = 0$ ; $7 + k -3k -3 = 0$

⇒ $4 - 2k = 0$ ; $4-2k=0$

⇒ $4 = 2k$ ;

⇒ $k = 2$ ;

⇒$1 : k = 1 : 2$

∴ The ratio in which the line $x + y = 4$ divides the line joining $A(-1,1)$ and $B(5,7)$ = $1 : 2$

Answer 2

Answer:

• Ratio of division = 1 : 2

Step-by-step explanation:

Let's assume the point of Intersection be (x, y) and also assume that the line x+y=4 divides the line joining the given points in the ratio λ : 1.

Now, we are using section formula to find the ratio of division.

Section formula:-

• $\sf(x,y) = \left( \dfrac{m_1x_2 + m_2x_1}{m_1 + m_2}, \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2} \right)$

Here, m1 : m2 = λ : 1.

By substituting the values in the formula, we get:

$\sf \implies(x,y) = \left( \dfrac{5 \lambda + ( - 1)}{ \lambda + 1}, \dfrac{7 \lambda + 1}{ \lambda + 1} \right)$

$\sf \implies(x,y) = \left( \dfrac{5 \lambda - 1}{ \lambda + 1}, \dfrac{7 \lambda + 1}{ \lambda + 1} \right)$

Now, since these coordinates are the point of Intersection, they must satisfy the given equation of line.

$\sf \implies x + y = 4$

$\sf \implies \dfrac{5 \lambda - 1}{ \lambda + 1} + \dfrac{7 \lambda + 1}{ \lambda + 1} = 4$

$\sf \implies \dfrac{5 \lambda - 1 + 7 \lambda + 1}{ \lambda + 1}= 4$

$\sf \implies \dfrac{12 \lambda }{ \lambda + 1}= 4$

$\sf \implies 3 \lambda = \lambda + 1$

$\sf \implies 2 \lambda = 1$

$\boxed{ \sf \implies \lambda = \dfrac{1}{2} }$

Therefore the ratio of division = λ : 1 = 1/2 : 1 or 1:2