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.Ex:

**various forms of ****straight**** lines:****General 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.As 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). **Parametric 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 y = 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 \((y-{{y}_{_{1}}})=\frac{({{y}_{2}}-{{y}_{1}})}{({{x}_{2}}-{{x}_{1}})}(x-{{x}_{1}})\)**Intercept 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.

\(\frac{y0}{xa}=\frac{0b}{a0}\),

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.**Normal 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.Consider 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.

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

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