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 intersects the x-axis where y = 0

Hence, the coordinates 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`

⇒ `k = 2/1`

Thus the required ratio is 2 : 1

iv. Slope of line AC:

m = `(-4 - 0)/(-2 - 4)`

= `(-4)/(-6)`

∴ m = `2/3`

Thus, the equation of the line AC is given by:

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

3y = 2x – 8

2x – 3y + 8 = 0 or 2x – 3y = 8