Subscribe to our YouTube channel for the latest videos, updates, and tips.


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.

 

3.  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\(_{1}\))].


4. 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).

 The Straight Line




11 and 12 Grade Math 

From Problems on Straight Lines to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Speed Distance and Time | Relation between Speed Distance and Time

    May 21, 25 12:58 PM

    Speed is defined as the distance covered per unit time. Speed = (Distance Travelled)/(Time Taken) Or, S = D/T. Speed also requires a unit of measurement. If the distance is in kilometres

    Read More

  2. Math Problem Answers | Solved Math Questions and Answers | Free Math

    May 21, 25 12:45 PM

    Partial fraction
    Math problem answers are solved here step-by-step to keep the explanation clear to the students. In Math-Only-Math you'll find abundant selection of all types of math questions for all the grades

    Read More

  3. Test of Divisibility | Divisibility Rules| Divisible by 2, 3, 5, 9, 10

    May 21, 25 10:29 AM

    The test of divisibility by a number ‘x’ is a short-cut method to detect whether a particular number ‘y’ is divisible by the number ‘x’ or not. Test of divisibility by 2: A number is divisible by 2

    Read More

  4. Divisible by 7 | Test of Divisibility by 7 |Rules of Divisibility by 7

    May 21, 25 10:17 AM

    Divisible by 7
    Divisible by 7 is discussed below: We need to double the last digit of the number and then subtract it from the remaining number. If the result is divisible by 7, then the original number will also be

    Read More

  5. Average Word Problems | Worksheet on Average Questions with Answers

    May 20, 25 05:40 PM

    In average word problems we will solve different types of problems on basic concept of average. 1. Richard scored 80, 53, 19, 77, 29 and 96 runs in 6 innings in a series. Find the average runs scored…

    Read More