# 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. 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.

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

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