Question

Given the equations of two straight lines, L1 and L2 are x – y = 1 and x + y = 5 respectively. If L1 and L2 intersects at point Q(3, 2). Find: 

1. the equation of line L3 which is parallel to L1 and has y-intercept 3.
2. the value of k, if the line L3 meets the line L2 at a point P(k, 4).
3. the coordinate of R and the ratio PQ : QR, if line L2 meets x-axis at point R.

Answer 1

a. L1: x – y = 1

⇒ y = x – 1

Comparing above equation with y = mx + c, we get:

⇒ m = 1.

We know that,

Slope of parallel lines are equal.

∴ Slope of L3 = 1.

Given,

L3 has y-intercept = 3.

∴ y = mx + c

⇒ y = 1.x + 3

⇒ y = x + 3.

Hence, equation of line L3: y = x + 3.

b. L2: x + y = 5 and L3: y = x + 3

⇒ x + y = 5  ...(1)

⇒ y = x + 3   ...(2)

Substituting the value of y from equation (2) in (1), we get:

⇒ x + (x + 3) = 5

⇒ 2x + 3 = 5

⇒ 2x = 5 – 3

⇒ 2x = 2

⇒ `x = 2/2`

⇒ x = 1.

Substituting value of x in equation (2), we get :

⇒ y = 1 + 3 = 4.

⇒ (x, y) = (1, 4)

∴ P(k, 4) = (1, 4)

Hence, value of k = 1.

c. Given,

L2 meets x-axis at point R.

At point on x-axis, y-coordinate = 0.

L2: x + y = 5

⇒ x + 0 = 5

⇒ x = 5

⇒ R = (x, y) = (5, 0).

P = (1, 4), Q = (3, 2) and R = (5, 0)

Let PQ : QR = k : 1

By section formula,

⇒ `(x, y) = ((m_1x_2 + m_2x_1)/(m_1 + m_2), (m_1y_2 + m_2y_1)/(m_1 + m_2))`

⇒ `(3, 2) = ((k xx 5 + 1 xx 1)/(k + 1), (k xx 0 + 1 xx 4)/(k + 1))`

⇒ `(3, 2) = ((5k + 1)/(k + 1), 4/(k + 1))`

⇒ `(5k + 1)/(k + 1) = 3` and `2 = 4/(k + 1)`

⇒ 5k + 1 = 3(k + 1) and 4 = 2(k + 1)

⇒ 5k + 1 = 3k + 3 and 4 = 2k + 2

⇒ 5k – 3k = 3 – 1 and 2k = 4 – 2

⇒ 2k = 2 and 2k = 2

⇒ `k = 2/2`

⇒ k = 1

∴ PQ : QR = 1 : 1.

Hence, R = (5, 0) and PQ : QR = 1 : 1.