 # Application of straight Lines

• The equation of a line parallel to ax + by + c = 0 is of the form ax + by + k = 0 , K ∈ R
• The equation of a line perpendicular to ax + by + c = 0 is of the form bx – ay + k = 0
• Equation of a line passing through (x1 y1) and parallel to ax + by + c = 0 is a (x – x1) + b (y – y1) = 0
• Equation of a line passing through (x1 y1) and perpendicular to ax + by + c = 0 is b (x – x1) − a (y – y1) = 0
• The image of the point P(x1, y1)) with respect to X-axis is Q(x1, − y1).
• The image of the point P(x1, y1) with respect to Y-axis is Q(− x1, y1).
• The image of the point P(x1, y1) with respect to mirror y = x is Q(y1, x1).
• The image of the point P(x1, y1) with respect to the origin is the point (− x1, −y1).
• The point (x1 y1) lies between the above parallel lines or does not lie between them according as \tt \frac{ax_{1}+by_{1}+c_{1}}{ax_{2}+by_{2}+c_{2}} is negative or positive.

### View the Topic in this video From 00:40 To 51:00

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1. Let the image of a point (x1, y1) with respect to ax + by + c = 0 be (x2, y2), then \tt \frac{x_{2}-x_{1}}{a}=\frac{y_{2}-y_{1}}{b}=\frac{-2\left(ax_{1}+by_{1}+c\right)}{a^{2}+b^{2}}

2. Point of intersection of two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is \tt \left(\frac{b_{1}c_{2}-b_{2}c_{1}}{a_{1}b_{2}-a_{2}b_{1}},\frac{c_{1}a_{2}-c_{2}a_{1}}{a_{1}b_{2}-a_{2}b_{1}}\right)

3. The length of perpendicular from a point (x1, y1) to a line ax + by + c = 0 is \tt \begin{vmatrix}\frac{ax_{1}+by_{1}+c}{\sqrt{a^{2}+b^{2}}} \end{vmatrix}
4. The foot of the perpendicular (h, k) from (x1, y1) to the line ax + by + c = 0 is given by \tt \frac{h-x_{1}}{a}=\frac{k-y_{1}}{b}=-\frac{\left(ax_{1}+by_{1}+c\right)}{a^{2}+b^{2}}.
5. (a). Foot of the perpendicular from (a, b) on x – y = 0 is \tt \left(\frac{a+b}{2},\frac{a+b}{2}\right).
(b). Foot of the perpendicular from (a, b) on x + y = 0 is \tt \left(\frac{a-b}{2},\frac{a-b}{2}\right).
6. The image of the line a1x + b1y + c1 = 0 about the line ax + by + c = 0 is 2(aa1 + bb1) (ax + by + c) = (a2 + b2) (a1x + b1y + c1).

7. The equation of a line parallel and lying midway between the above two lines is ax + by + \tt \frac{c_{1}+c_{2}}{2}=0

8. If (h , k) is the foot of the perpendicular from (x1 y1) to the line ax + by + c = 0 then \tt \frac{h-x_{1}}{a}= \frac{k-y_{1}}{b}=\frac{-\left(ax_{1}+by_{1}+c\right)}{a^{2}+b^{2}}

9. If (h , k) is the image (reflection) of the point (x1 y1) w.r.t the line ax + by + c = 0 then \tt \frac{h-x_{1}}{a}= \frac{k-y_{1}}{b}=\frac{-2\left(ax_{1}+by_{1}+c\right)}{a^{2}+b^{2}}

10. If ‘B’ is image of ‘A’ w.r.t ‘P’ then 2P = A + B.