ICSE Class 10 Maths Chapter 14 Equation of a Line

Equation of a Line

A Basic Concept

In class IX, in the chapter of graphs, students have drawn lines for the given equations, like: \(3x - 2y = 7\), \(x + 5y = 8\), \(x = 5\), \(y + 3 = 0\). Such equations, which are of first degree in variables x and y both or only in x or only in y, are known as linear equations and each equation always represents a straight line.

In other words, every straight line can be represented by a linear equation.

Remember: (i) Any point, which satisfies the equation of a line, lies on that line. (ii) Any point, through which a line passes, will always satisfy the equation of that line.

Teacher's Note

Linear equations describe real-world relationships like the cost of items based on quantity, making abstract math concrete and practical for daily calculations.

Example 1

(i) Check, whether point \((4, -2)\) lies on the line represented by equation \(3x + 5y = 2\) or not?

(ii) The straight line represented by equation \(x - 3y + 8 = 0\) passes through \((2, 4)\). Is this true?

Solution

(i) Substituting \(x = 4\) and \(y = -2\) in the given equation, we get: \(3 \times 4 + 5 \times -2 = 2\) \(\Rightarrow\) \(12 - 10 = 2\), which is true.

Therefore, Point \((4, -2)\) satisfies the given equation and so it lies on the line represented by the equation \(3x + 5y = 2\).

(ii) Substituting \(x = 2\) and \(y = 4\) in the given equation, we get: \(2 - 3 \times 4 + 8 = 0\) \(\Rightarrow\) \(2 - 12 + 8 = 0\), which is not true.

Therefore, The line, represented by the equation \(x - 3y + 8 = 0\), does not pass through the point \((2, 4)\).

Example 2

The line, represented by the equation \(3x - 8y = 2\), passes through the point \((k, 2)\). Find the value of k.

Solution

Substituting \(x = k\) and \(y = 2\) in the given equation, we get: \(3k - 8 \times 2 = 2\) \(\Rightarrow\) \(3k - 16 = 2\) i.e. \(k = 6\).

Example 3

Does the line \(3x = y + 1\) bisect the line segment joining A \((-2, 3)\) and B \((4, 1)\)?

Solution

The given line will bisect the join of AB, if the co-ordinates of the mid-point of AB satisfy the equation of the line.

Mid-point of A \((-2, 3)\) and B \((4, 1)\)

\[= \left(\frac{-2 + 4}{2}, \frac{3 + 1}{2}\right) = (1, 2)\]

Substituting \(x = 1\) and \(y = 2\) in the given equation, we get: \(3 \times 1 = 2 + 1\) \(\Rightarrow\) \(3 = 3\), which is true.

Therefore, The given line bisects the join of A and B.

Teacher's Note

Understanding midpoints and line bisectors helps students grasp how geometric properties are verified algebraically, bridging abstract theory with measurable outcomes.

Exercise 14(A)

1. Find, which of the following points lie on the line \(x - 2y + 5 = 0\):

(i) \((1, 3)\) (ii) \((0, 5)\) (iii) \((-5, 0)\) (iv) \((5, 5)\) (v) \((2, -1.5)\) (vi) \((-2, -1.5)\)

2. State, true or false:

(i) the line \(\frac{x}{2} + \frac{y}{3} = 0\) passes through the point \((2, 3)\)

(ii) the line \(\frac{x}{2} + \frac{y}{3} = 0\) passes through the point \((4, -6)\)

(iii) the point \((8, 7)\) lies on the line \(y - 7 = 0\)

(iv) the point \((-3, 0)\) lies on the line \(x + 3 = 0\)

(v) if the point \((2, a)\) lies on the line \(2x - y = 3\), then \(a = 5\)

3. The line given by the equation \(2x - \frac{y}{3} = 7\) passes through the point \((k, 6)\); calculate the value of k.

4. For what value of k will the point \((3, -k)\) lie on the line \(9x + 4y = 3\)?

5. The line \(\frac{3x}{2} - \frac{2y}{3} + 1 = 0\) contains the point \((m, 2m - 1)\); calculate the value of m.

6. Does the line \(3x - 5y = 6\) bisect the join of \((5, -2)\) and \((-1, 2)\)?

7. (i) The line \(y = 3x - 2\) bisects the join of \((a, 3)\) and \((2, -5)\), find the value of a. (ii) The line \(x - 6y + 11 = 0\) bisects the join of \((8, -1)\) and \((0, k)\). Find the value of k.

8. (i) The point \((-3, 2)\) lies on the line \(ax + 3y + 6 = 0\), calculate the value of a. (ii) The line \(y = mx + 8\) contains the point \((-4, 4)\), calculate the value of m.

9. The point P divides the join of \((2, 1)\) and \((-3, 6)\) in the ratio 2 : 3. Does P lie on the line \(x - 5y + 15 = 0\)?

10. The line segment joining the points \((5, -4)\) and \((2, 2)\) is divided by the point Q in the ratio 1 : 2. Does the line \(x - 2y = 0\) contain Q?

11. Find the point of intersection of the lines \(4x + 3y = 1\) and \(3x - y + 9 = 0\). If this point lies on the line \((2k - 1)x - 2y = 4\); find the value of k.

The above question can also be stated as: If the lines \(4x + 3y = 1\), \(3x - y + 9 = 0\) and \((2k - 1)x - 2y = 4\) are concurrent (pass through the same point), find the value of k.

12. Show that the lines \(2x + 5y = 1\), \(x - 3y = 6\) and \(x + 5y + 2 = 0\) are concurrent.

Inclination of a Line

The inclination of a line is the angle \(\theta\) which the part of the line (above x-axis) makes with x-axis.

If measured in anti-clockwise direction the inclination \(\theta\) is positive and if measured in clockwise direction, the inclination \(\theta\) is negative.

1. Inclination \((\theta)\) of the x-axis and every line parallel to the x-axis is \(0°\).

2. Inclination \((\theta)\) of the y-axis and every line parallel to the y-axis is \(90°\).

Teacher's Note

Understanding inclination helps students visualize how angles relate to line slopes, making it easier to graph functions and predict line behavior in coordinate geometry.

Concept of Slope (or, gradient)

The slope of any inclined plane is the ratio between the vertical rise of the plane and its horizontal distance.

i.e slope of AC = \(\frac{\text{vertical rise}}{\text{horizontal distance}} = \frac{AB}{BC} = \tan \theta\) where \(\theta\) is the angle which the plane makes with the horizontal

Slope (or, gradient) of a Straight Line

The slope of a straight line is the tangent of its inclination and is denoted by letter m.

i.e if the inclination of a line is \(\theta\), its slope \(m = \tan \theta\).

1. Slope of the x-axis is \(m = \tan 0° = 0\) (Since, \(\theta = 0°\))

2. Slope of the y-axis is \(m = \tan 90° = \infty\) i.e. not defined (Since, \(\theta = 90°\))

3. The slope of a line is positive, if it makes an acute angle in the anti-clockwise direction with x-axis.

4. The slope of a line is negative, if it makes an obtuse angle in the anti-clockwise direction with the x-axis or an acute angle in the clockwise direction with the x-axis.

For example: (i) Inclination \(\theta = 45°\) - Slope = \(\tan 45° = 1\)

(ii) Inclination \(\theta = 135°\) or \(-45°\) - Slope = \(\tan(-45°) = -\tan 45° = -1\)

Teacher's Note

Understanding slope as a measure of steepness helps students comprehend how quickly a line rises or falls, essential for analyzing trends in data and physical phenomena.

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