Straight Lines

Straight line is the locus of a moving point P (h, k), which moves in such a condition that P is always collinear with the given two fixed points.straight-linesEx:various-forms-of-straight-lines-1

various forms of straight lines:various-forms-of-straight-linesGeneral form: The most general equation of a straight line is ax + by + c = 0, where a, b and c are any real numbers such that both a and b can’t be zero simultaneously.

Slope intercept form: If we have a straight line whose slope is ‘m’ and which makes an intercept ‘c’ on the y-axis then its equation is given by y = mx + c.slope-intercept-formAs shown in the figure above, the y-intercept here is c.

Slope one point form: The equation of a straight line having slope as ‘m’ and which passes through the point (x₁, y₁) is given by (y-y₁) = m(x-x₁).  slope-one-point-formParametric form: Consider line PQ with coordinates P(x, y) and Q(x₁, y₁). Then Co-ordinates of any points P(x, y) are

x = x₁ + r cos θ parametric-formy = y₁ + r sin θ

Equation of the line is obtained as follows: –

⇒ \(\frac{xx}{cos~\theta }=\frac{\text{ }yy}{sin~\theta }=r\)

This is parametric form of the equation of a straight line.

Two points form: If we have two given points say (x₁, y₁) and (x₂, y₂), then the line passing through them is given by the formula

\((y-{{y}_{1}})\text{ }=\text{ }m(x-{{x}_{1}})\) or \((y-{{y}_{_{1}}})=\frac{({{y}_{2}}-{{y}_{1}})}{({{x}_{2}}-{{x}_{1}})}(x-{{x}_{1}})\)two-points-formIntercept form: If intercepts of a line on x and y-axis are known then equation of the line can also be found in two-intercept form. Intercepts are OA and OB on x and y-axis respectively, where A(a, 0) and B(0, b) are two points through which line is passing.


where P(x, y) is any point on the line

If we are given the intercepts of a line on the x and y axis respectively as ‘a’ and ‘b’ then the equation of the straight line is given by \(\frac{x}{a}+\frac{y}{b}=1\).

⇒ \(\frac{y}{b}\text{ }=\text{ }\frac{x}{a}\text{ }+\text{ }1\).

⇒ \(\frac{x}{a}+\frac{y}{b}=1\).

This is intercept from of the equation of a straight line.intercept-formNormal form: x cos α + y sin α = a is the equation of the straight line in perpendicular form, where ‘p’ is the length of the perpendicular from the origin O on the line and this perpendicular makes an angle α with the positive direction of x-axis.normal-formConsider line l as shown in figure given above

ON ⊥ l and |ON| = p

We have in triangle ONA

\(OA=\frac{p}{cos~\alpha }\)

A and B are intercept points of line l. So intercepts on x and y-axes are p/cos α and p/sin α respectively. So equation of line.

\(\frac{x\text{ }cos~\alpha }{p}+\frac{y\text{ }sin~\alpha }{p~}=\text{ }1\).

⇒ \(x\text{ }cos~\alpha ~+\text{ }y\text{ }sin~\alpha ~=\text{ }p\).