Problems on Straight Lines

We will learn how to solve different types of problems on straight lines.

1. Find the angle which the straight line perpendicular to the straight line √3x + y = 1, makes with the positive direction of the x-axis.

Solution:   

The given equation of the straight line √3x + y = 1

Covert the above equation into slope-intercept form we get,

y = - √3x + 1…………………… (i)

Let us assume that the given straight line (i) makes an angle θ with the positive direction of the x-axis.

Then, the slope of the straight line (i) will be tan θ

Hence, we must have, tan  = - √3 [Since, the slope of the straight line y = - √3x + 1 is - √3]

⇒ tan θ = - tan 60° = tan (180° - 60°) = tan 120°

⇒ tan θ = 120°

Since the straight line (i) makes an angle 120° with the positive direction of the x-axis, hence a straight line perpendicular to the line (i) will make an angle 120° - 90° = 30° with the positive direction of the x-axis.

2. Prove that P (4, 3), Q (6, 4), R (5, 6) and S (3, 5) are the angular points of a square.

Solution:

We have,

PQ = \(\sqrt{(6 - 4)^{2} + (4 - 3)^{2}}\) = √5

QR = \(\sqrt{(6 - 4)^{2} + (5 - 4)^{2}}\) = √5

RS = \(\sqrt{(5 - 6)^{2} + (3 - 5)^{2}}\) = √5 and

SP = \(\sqrt{(5 - 3)^{2} + (3 - 4)^{2}}\) = √5

Therefore, PQ = QR = RS = SP.

Now, m\(_{1}\) = Slope of PQ = \(\frac{4 - 3}{6 - 4}\) = ½

m\(_{2}\) = Slope of QR = \(\frac{6 - 4}{5 - 6}\) = -2 and

m\(_{3}\) =  Slope of RS = \(\frac{5 - 6}{3 - 5}\) = ½

Clearly, m\(_{1}\) ∙ m\(_{2}\) = ½ ∙ (-2) = -1 and m\(_{1}\) = m\(_{3}\).

This shows that PQ is perpendicular to QR and PQ is parallel to RS.

Thus, PQ = QR = RS = SP, PQ ⊥ QR and PQ is parallel to RS.

Thence, PQRS is a square.

 

2.  A straight line passes through the point (- 1, 4) and makes an angle 60° with the positive direction of the x-axis. Find the equation of the straight line.

Solution:  

The required line makes an angle 60° with the positive direction of the axis of x.

Therefore, the slope of the required line = m = tan 60° = √3. Again, the required line passes through the point (- 1, 4).

Therefore, the equation of the required straight line is

y - 4 = √3(x + 1), [Using the point-slope form, y - y\(_{1}\)  = m (x - x\(_{1}\))].

3. Find the equation of the straight line which passes through the point (5, 6) and has intercepts on the axes equal in magnitude but opposite in sign. Find also the co-ordinates of the point on the line at which the ordinate is double the abscissa.

Solution:    

Let us assume that, the equation of the required straight line be

\(\frac{x}{a}\) + \(\frac{y }{b}\) = 1 ………………. (i)

According to the question, b = - a; hence, equation (i) reduces to

\(\frac{x}{a}\) + \(\frac{y }{-a}\) = 1

⇒ x - y = a ………………. (ii)

Again, the line (ii) passes through the point (5, 6). Therefore,

5 - 6 = a       

⇒ a = - 1

Therefore, the equation of the required straight line is,

x- y = -1     

⇒ x- y + 1 = 0………………. (iii)

Now, we are to find the co-ordinates of that point on the line (iii) for which the ordinate is double the abscissa.

Let the co-ordinates of the required point be (α, β). Then the point (α, β) will satisfy the equation (iii).

Therefore, α - 2α + 1 = 0     

⇒ α = 1.

Therefore, the co-ordinates of the required point are (1, 2).

11 and 12 Grade Math 

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