#### Question

In the given figure 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 coordinates of A.

2) Find the length of AB and AC

3) Find the ratio in which Q divides AC.

4) Find the equation of the line AC

#### Solution

1) The line intersects the x-axis where, y = 0. Thus, the coordinates of A are (4, 0).

2) Length of AB = `sqrt((4-(-2))^2+ (0 -3)^2) = sqrt(36 + 9) = sqrt45 = 3sqrt5`units

3) Let Q divides AC in the ratio m1 : m2. Thus, the co-ordinates of Q are (0, y)

Since x = `(m_1x_2 + m_2x_1)/(m_1 + m_2)`

`=> 0 = (m_1 (-2) + m_2(4))/(m_1 + m_2) => 2m_1 = 4m_2 => m_1 = 2m_2`

`=> m_1/m_2 = 2/1`

∴ Required ratio is 2 : 1

4) A(4, 0) = A(x1,y1) and B(-2,-4) = b(x2, y2)

Slope of AC = `(-4-0)/(-2-4) = (-4)/(-6) = 2/3`

∴ Equation of line AC is given by `y - y_1 = m(x - x_1)`

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

`=> 3y = 2x - 8`

`=> 2x - 3y = 8`