Learn Equation of Straight Lines | How to Calculate Slope of Line

A relation between x and y which is satisfied by co-ordinates of every point lying on a line is called equation of the straight line. Here, remember that every one degree equation in variable x and y always represent a straight line i.e. ax + by + c = 0 ; a & b \(\ne\) 0 simultaneously. Later you get to know how to calculate slope of line, angle between two lines formula and different equation of straight lines.

(a)   Equation of straight lines parallel to x-axis at a distance 'a' is y = a or y = -a.
(b)   Equation of x-axis is y = 0
(c)   Equation of straight lines parallel to y-axis at a distance 'b' is x = b or x = -b.
(d)   Equation of y-axis is x = 0.

Angle between two lines formula :

(a)   If \(\theta\) be the angle between two lines : y = \(m_1x + c_1\) and y = \(m_2x + c_2\), then

tan\(\theta\) = \(\pm\) (\(m_1-m_2\over {1+m_1m_2}\))

(b)   If the equation of lines are \(a_1x + b_1y + c_1\) = 0 and \(a_2x + b_2y + c_2\) = 0, then these lines are -
      (i)   Parallel   \(\iff\)   \(a_1\over a_2\) = \(b_1\over b_2\) \(\ne\) \(c_1\over c_2\)

      (ii)   Perpendicular   \(\iff\)   \(a_1a_2\) + \(b_1b_2\) = 0

      (iii)   Coincident   \(\iff\)   \(a_1\over a_2\) = \(b_1\over b_2\) = \(c_1\over c_2\)

How to Calculate Slope of Line :

If a given line makes an angle \(\theta\) (0 \(\le\) \(\theta\) \(\le\) 180 , \(\theta\) \(\ne\) 90) with the positive direction of x-axis, then the slope of this line will be tan\(\theta\) and is usually denoted by letter m i.e. m = tan\(\theta\). If A(\(x_1,y_1\)) and B(\(x_2,y_2\)) & \(x_1\) \(\ne\) \(x_2\) then the slope of line m = \(y_2-y_1\over {x_2-x_1}\)
Remarks :
(i)   If \(\theta\) = 90, m does not exist and line is parallel to y-axis.
(ii)   If \(\theta\) = 0, m = 0 and the line is parallel to x-axis.
(iii)   Let \(m_1\) and \(m_2\) be slopes of two given lines(none of them is perpendicular to y-axis)
        (a)   If lines are parallel, \(m_1\) = \(m_2\) and vice-versa.
        (b)  If lines are perpendicular, \(m_1\)\(m_2\) = -1 and vice-versa.

Different Equation of Straight Lines - formula

(a)Equation of Straight Lines in Slope Intercept form :

Let m be the slope of a line and c its intercept on y-axis. Then the equation of this straight line is written as y = mx + c.

(b)Equation of Straight Lines in Point Slope form :

Let m be the slope of a line and it passes through a point (\(x_1,y_1\)), then its equation is written as : y - \(y_1\) = m(x - \(x_1\))

(c) Equation of Straight Lines in Two Point form :

Equation of a line passing through two points (\(x_1,y_1\)) and (\(x_2,y_2\)) is written as

y - \(y_1\) = \(y_2-y_1\over {x_2-x_1}\)(x - \(x_1\))

(d) Equation of Straight Lines in Intercept form :

If a and b are the intercepts made by a line on the axes of x and y, its equation is written as : \(x\over a\) + \(y\over b\) = 1
(i)   Length of intercept of line between the coordinate axes = \(\sqrt{a^2+b^2}\)

(e) Equation of Straight Lines in Normal form :

If p is the length of perpendicular on a line from the origin, and \(\alpha\) the angle which this perpendicular makes with positive x-axis, then the equation of this line is written as xcos\(\alpha\) + ysin\(\alpha\) = p(p is always positive)   where 0 \(\le\) \(\alpha\) < 2\(\pi\).

(f) Equation of Straight Lines in Parametric form :

To find the equation of a straight line which passes through a given point A(h,k) and makes a given angle \(\theta\) with the positive direction of the axis. P(x,y) is any point on the line.
Let AP = r, then x - h = rcos\(\theta\), y - k = rsin\(\theta\) & \(x - h\over {cos\theta}\) = \(y - k\over {sin\theta}\) = r is the equation of straight line.

(g)Equation of Straight Lines in General form :

We know that a first degree equation in x and y, ax + by + c = 0 always represent a straight line. This form is known as general form of straight line.
(i)   Slope of this line = -\(a\over b\)
(ii)   Intercept by this line on x-axis = -\(c\over a\) and Intercept by this line on y-axis = -\(c\over b\)
(iii)   To change the general form of a line to normal form, first take c to right hand side and make it positive, then divide the whole equation by \(\sqrt{a^2+b^2}\).

Example : Equation of a line which passes through point A(2,3) and makes an angle of 45 with x-axis. If this line meet the line x + y + 1 = 0 at a point P then distance AP is-

Solution : Here \(x_1\) = 2, \(y_1\) = 3 and \(\theta\) = 45

Hence \(x-2\over cos45\) = \(y-3\over sin45\) = r

from first two parts \(\implies\) x - 2 = y - 3 \(\implies\) x - y + 1 = 0

Co-ordinate of point P on this line is (2+\(r\over \sqrt{2}\), 3+\(r\over \sqrt{2}\)).

If this point is on line x + y + 1 = 0 then

(2+\(r\over \sqrt{2}\)) + (3+\(r\over \sqrt{2}\)) + 1 = 0     \(\implies\) r = -\(3\sqrt{2}\) ; |r| = \(3\sqrt{2}\)





Equation of Straight lines parallel and perpendicular to a given line :

(a)   Equation of line parallel to line ax + by + c = 0 is ax + by + \(\lambda\)
(b)   Equation of line perpendicular to line ax + by + c = 0 is bx - ay + k
here \(\lambda\), k are parameters and their values are obtained with the help of additional information given in the problem

Equation of Straight lines making an angle with a line :

Equation of line passing through a point (\(x_1,y_1\)) and making an angle \(\alpha\), with the line y = mx + c is written as

y - \(y_1\) = \(m \pm tan\alpha\over {1 \mp mtan\alpha}\)(x - \(x_1\))

Length of perpendicular from a point on a line :

Length of perpendicular from a point (\(x_1,y_1\)) on the line ax + by + c = 0 is |\(ax_1 + by_1 + c\over {\sqrt{a^2+b^2}}\)|

In particular, the length of perpendicular from the origin on the line ax + by + c = 0 is

P = \(|c|\over {\sqrt{a^2+b^2}}\)

Distance between two parallel lines :

(a)   The distance between two parallel lines ax + by + \(c_1\) = 0 and ax + by + \(c_2\) = 0 is

d = \(|c_1-c_2|\over {\sqrt{a^2+b^2}}\)

(b)   The area of parallelogram = \(p_1p_2\over sin\theta\), where \(p_1\) & \(p_2\) are distances between two pairs of opposite sides & \(\theta\) is the angle between any two adjacent sides.

Concurrency of lines :

(a)   Three lines \(a_1x + b_1y + c_1\) = 0 and \(a_2x + b_2y + c_2\) = 0 and \(a_3x + b_3y + c_3\) = 0 are concurrent,

if \(\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}\) = 0

(b)   To test the concurrency of three lines, first find out the point of interesection of the three lines. If this point lies on the remaining line (i.e. coordinates of the point satisfy the equation of the line) then the three lines are concurrent otherwise not concurrent.

Reflection and image of a point in a line :

Let P(x,y) be any point, then its image with respect to

(a)   x-axis is Q(x,-y)
(b)   y-axis is R(-x,y)
(c)   origin is S(-x,-y)
(d)   line y = x is T(y,x)
(e)   Reflection of a point about any arbitrary line : The image (h,k) of a point P(\(x_1,y_1\)) about the line ax+by+c = 0 is given by following formula.

\(h-x_1\over a\) = \(k-y_1\over b\) = -2(\(ax_1+by_1+c\over {a^2+b^2}\))

and the foot of perpendicular (p,q) from a point (\(x_1,y_1\)) on the line ax+by+c = 0 is given by following formula.

\(p-x_1\over a\) = \(q-y_1\over b\) = -(\(ax_1+by_1+c\over {a^2+b^2}\))


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