# Word Problems on Straight Lines

Here we will solve different types of word problems on straight lines.

1.Find the equation of a straight line that has y-intercept 4 and is perpendicular to straight line joining (2, -3) and (4, 2).

Solution:

Let m be the slope of the required straight line.

Since the required straight line is perpendicular to the line joining P (2, -3) and Q (4, 2).

Therefore,

m × Slope of PQ = -1

⇒ m ×  $$\frac{2 + 3}{4 - 2}$$ = -1

⇒ m ×  $$\frac{5}{2}$$ = -1

⇒ m = -$$\frac{2}{5}$$

The required straight lien cut off an intercept of length 4 on y-axis.

Therefore, b = 4

Hence, the equation of the required straight line is y = -$$\frac{2}{5}$$x + 4

⇒ 2x + 5y - 20 = 0

2. Find the co-ordinates of, the middle point of the portion of the line 5x + y = 10 intercepted between the x and y-axes.

Solution:

The intercept form of the given equation of the straight line is,

5x + y = 10

Now dividing both sides by 10 we get,

⇒ $$\frac{5x}{10}$$+ $$\frac{y}{10}$$ = 1

⇒ $$\frac{x}{2}$$ + $$\frac{y}{10}$$ = 1.

Therefore, it is evident that the given straight line intersects the x-axis at P (2, 0) and the y-axis at Q (0, 10).

Therefore, the required co-ordinates of the middle point of the portion of the given line intercepted between the co-ordinate axes = the co-ordinates of the middle point of the line-segment PQ

= ($$\frac{2 + 0}{2}$$, $$\frac{0 + 10}{2}$$)

= ($$\frac{2}{2}$$, $$\frac{10}{2}$$)

= (1, 5)

More examples on word problems on straight lines.

3. Find the area of the triangle formed by the axes of co-ordinates and the straight line 5x + 7y = 35.

Solution:

The given straight line is 5x + 7y = 35.

The intercept form of the given straight line is,

5x + 7y = 35

⇒ $$\frac{5x}{35}$$+ $$\frac{7y}{35}$$ = 1, [Dividing both sides by 35]

⇒ $$\frac{x}{7}$$ + $$\frac{y}{5}$$ = 1.

Therefore, it is evident that the given straight line intersects the x-axis at P (7, 0) and the y-axis at Q (0, 5).

Thus, if o be the origin then, OP = 7 and OQ = 5

Therefore, the area of the triangle formed by the axes of co-ordinates and the given line = area of the right-angled ∆OPQ

= ½ |OP × OQ|= ½ ∙ 7 . 5 = $$\frac{35}{2}$$ square units.

4. Prove that the points (5, 1), (1, -1) and (11, 4) are collinear. Also find the equation of the straight line on which these points lie.

Solution:

Let the given points be P (5, 1), Q (1, -1) and R (11, 4). Then the equation of the line passing through P and Q is

y - 1 = $$\frac{-1 - 1}{1 - 5}$$(x - 5)

⇒ y - 1 = $$\frac{-2}{-4}$$(x - 5)

⇒ y - 1 = $$\frac{1}{2}$$(x - 5)

⇒ 2(y - 1) = (x - 5)

⇒ 2y - 2 = x - 5

⇒ x - 2y - 3 = 0

Clearly, the point R (11, 4) satisfies the equation x - 2y - 3 = 0. Hence the given points lie on the same straight line, whose equation is x - 2y - 3 = 0.