## Straight Lines

# 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 (x
_{1}y_{1}) and parallel to ax + by + c = 0 is a (x – x_{1}) + b (y – y_{1}) = 0 - Equation of a line passing through (x
_{1}y_{1}) and perpendicular to ax + by + c = 0 is b (x – x_{1}) − a (y – y_{1}) = 0 - The image of the point P(x
_{1}, y_{1})) with respect to X-axis is Q(x_{1}, − y_{1}). - The image of the point P(x
_{1}, y_{1}) with respect to Y-axis is Q(− x_{1}, y_{1}). - The image of the point P(x
_{1}, y_{1}) with respect to mirror y = x is Q(y_{1}, x_{1}). - The image of the point P(x
_{1}, y_{1}) with respect to the origin is the point (− x_{1}, −y_{1}). - The point (x
_{1}y_{1}) 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 (x_{1}, y_{1}) with respect to ax + by + c = 0 be (x_{2}, y_{2}), 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 a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 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 (x_{1}, y_{1}) 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 (x_{1}, y_{1}) 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 a_{1}x + b_{1}y + c_{1} = 0 about the line ax + by + c = 0 is 2(aa_{1} + bb_{1}) (ax + by + c) = (a^{2} + b^{2}) (a_{1}x + b_{1}y + c_{1}).

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 (x_{1} y_{1}) 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 (x_{1} y_{1}) 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.