Find the ratio in which y axis divides the loline segment joining the

1.	Find the ratio in which y axis divides the loline segment joining the point A (5,-6),B (1,-4)
Answer» Given Line segment joining two points (5,-6) and (1,-4) is DIVIDED by y-axis. To Find: RATIO in which y-axis divides the line segment joining the points (5,-6) and (1,- 4) Sectional Formula $\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){$P$}\put(11.7,5.9){$Q$}\put(5.4,5.5){$(x_1\,,\,y_1)$}\put(11.4,5.5){$(x_2\,,\,y_2)$}\put(8,6){\circle{0.2}}\put(7.8,6.3){$R$}\put(7.4,5.5){$(x\,,\,y)$}\put(6.6,6.3){$m$}\put(9.3,6.3){$n$}\put(11.7,5.9){$Q$}\end{picture}$ Let P(x₁,y₁) and Q(x₂,y₂) be two points. Let the point R(x,y) divide the line segment joining the points P and Q internally in the ratio m:n, then $\sf{(x,y)=\bigg(\dfrac{mx_{2}+nx_{1}}{m+n},\dfrac{my_{2}+ny_{1}}{m+n}\bigg)}$ Solution:* Let the y-axis divides the given line segment in λ:1 and point of intersection of y-axis and given line segment be (0,b) (0,b) because on y-axis, x-coordinate is ALWAYS 0 and y-coordinate can be any constant(say b). Diagram: $\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){$P$}\put(11.7,5.9){$Q$}\put(5.4,5.5){$(5\,,\,-6)$}\put(11.4,5.5){$(1\,,\,-4)$}\put(8,6){\circle{0.2}}\put(7.8,6.3){$R$}\put(7.4,5.5){$(0\,,\,b)$}\put(6.6,6.3){$\lambda$}\put(9.3,6.3){$1$}\put(11.7,5.9){$Q$}\end{picture}$ Now, by using sectional formula, we GET $\longrightarrow\sf{(0,b)=\bigg(\dfrac{\lambda(1)+5}{\lambda+1},\dfrac{\lambda(-4) -6}{\lambda +1}\bigg)}$ $\longrightarrow\sf{(0,b)=\bigg(\dfrac{\lambda+5}{\lambda+1},\dfrac{-4\lambda-6}{\lambda +1}\bigg)}$ On comparing x-coordinate of both the SIDES we get, $\sf{\dfrac{\lambda+5}{\:\:\:\lambda+1}=0}$ -----------(1) From (1), we get ✏ λ=-5 So, y-axis divides the given line segment in 5:1 externally Hence, the required ratio is 5:1*

Find the ratio in which y axis divides the loline segment joining the point A (5,-6),B (1,-4)

Answer»

Given

Line segment joining two points (5,-6) and (1,-4) is DIVIDED by y-axis.

To Find:

RATIO in which y-axis divides the line segment joining the points (5,-6) and (1,- 4)

Sectional Formula

$\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){$P$}\put(11.7,5.9){$Q$}\put(5.4,5.5){$(x_1\,,\,y_1)$}\put(11.4,5.5){$(x_2\,,\,y_2)$}\put(8,6){\circle*{0.2}}\put(7.8,6.3){$R$}\put(7.4,5.5){$(x\,,\,y)$}\put(6.6,6.3){$m$}\put(9.3,6.3){$n$}\put(11.7,5.9){$Q$}\end{picture}$

Let P(x₁,y₁) and Q(x₂,y₂) be two points. Let the point R(x,y) divide the line segment joining the points P and Q internally in the ratio m:n, then

$\sf{(x,y)=\bigg(\dfrac{mx_{2}+nx_{1}}{m+n},\dfrac{my_{2}+ny_{1}}{m+n}\bigg)}$

Solution:

Let the y-axis divides the given line segment in λ:1 and point of intersection of y-axis and given line segment be (0,b)

(0,b) because on y-axis, x-coordinate is ALWAYS 0 and y-coordinate can be any constant(say b).

Diagram:

$\setlength{\unitlength}{0.9 cm}}\begin{picture}(12,4)\thicklines\put(6,6){\line(1,0){5.5}}\put(5.6,5.9){$P$}\put(11.7,5.9){$Q$}\put(5.4,5.5){$(5\,,\,-6)$}\put(11.4,5.5){$(1\,,\,-4)$}\put(8,6){\circle*{0.2}}\put(7.8,6.3){$R$}\put(7.4,5.5){$(0\,,\,b)$}\put(6.6,6.3){$\lambda$}\put(9.3,6.3){$1$}\put(11.7,5.9){$Q$}\end{picture}$