Question

In the figure, given, ABC is a triangle and BC is parallel to the y-axis. AB and AC intersect the y-axis at P and Q respectively. 

1. Write the co-ordinates of A. 
2. Find the length of AB and AC.
3. Find the radio in which Q divides AC. 
4. Find the equation of the line AC.

Answer 1

i. The line intresects the x-axis where y = 0

Hence, the co-ordinates of A are (4, 0)

ii. Length of AB = `sqrt((-2 -4)^2 + (3 - 0)^2`

= `sqrt(36 + 9)`

= `sqrt(45)`

= `3sqrt(5)` units

Length of AC = `sqrt((-2 -4)^2 + (-4 - 0)^2`

= `sqrt(36 + 16)`

= `sqrt(52)`

= `2sqrt(13)` units

iii. Let K be the required ratio which divides the line segment joining the co-ordinates A(4, 0) and O(–2, −4)

Let the co-ordinates of Q be x and y 

∴ `x = (k(-2) + 1(4))/(k + 1)` and `y = (k(-4) + 0)/(k + 1)`

Q lies on the y-axis where x = 0,

`=> (-2k + 4)/(k + 1) = 0`

`=>` –2k + 4 = 0

`=>` 2k = 4

`=> k = 4/2 = 2/1`

Thus the required ratio is 2 : 1

iv. Slope of line AC = m = `(-4 - 0)/(-2 - 4) = (-4)/(-6) = 2/3`

Thus, the equation of the line AC is given by 

`y - 0 = 2/3(x - 4)`

i.e 3y = 2x – 8

i.e 2x – 3y = 8