Mathematics STRAIGHT LINE

The image or reflection of a point with respect to `y-` axis:

Let `P (alpha, beta)` be any point and `Q (x, y)` be its image about `y -` axis, then `( M` is the mid point of `PQ )`

`x= -alpha` and `y= beta`

`Q equiv(- alpha , beta)`

i.e sign change of abscissae.

`text(Funda :)` The image of the line `ax + by + c = 0` about y-axis is `- ax + by + c = 0`

The image or reflection of a point with respect to origin :

Let `P (alpha, beta)` be any point and `Q (x, y)` be its image about the origin

`(O` is the mid point of `PQ),` then

`x = - alpha ` and `y =- beta`

`Q equiv (- alpha , - beta)`

i.e , sign change of abscissae and ordinate.

`text(Funda :)` The image of the line `ax + by + c = 0` about origin is `- ax - by + c = 0.`

The image or reflection of a point with respect to the line `x = a :`

Let `P (alpha,beta)` .be any point and `Q (x, y)` be its image about the line `x = a`, then `y = beta`

Co-ordinates of `M` are `(a, beta)`

`Q =(2a-alpha, beta)`

`text(Funda :)` The image of the line `ax + by + c = 0` about the line `x = lambda` is `a (2 lambda-x) + by + c = 0`

The image or reflection of a point with respect to the line `y = b :`

Let `P (alpha, beta)` be any point and `Q (x, y)` be its image about the line `y = b` , then `x = a`

Co-ordinates of `M` are `(alpha, b)`

` M` is the mid point of `PQ`

`Q =(alpha, 2b-beta)`

`text(Funda :)` The image of the line `ax + by + c = 0` about the line `y =mu` is `ax+ b (2 mu - y) + c == 0`.

The image or reflection of a point with respect to the line `y = x :`

Let `P (alpha, beta)` be any point and `Q (x_1 , y_1)` be its image about the line `y = x (RS)` , then `PQ` `bot` `RS`

` ( `Slope of `PQ ) xx ( `Slope of `RS ) = -1`

`(y_1-beta)/(x_1 - alpha) xx 1 =-1`

` x_1 - alpha = beta - y_1`...............................`(1)`

and mid point of `PQ` lie on `y = x`

`((y_1+beta)/2) = ((x_1 +alpha)/2)`

`x_1 + alpha = beta +y_1`........................`(2)`

Solving `(1)` and `(2),` we get `x_1 = beta` and `y_1 = alpha`

`Q equiv (beta,alpha)`

i.e., interchange of `x` and `y.`

`text(Funda :)` The image of the line `ax + by + c = 0` about the line `y = x` is `ay + bx + c = 0` .

The image or reflection of a point with� respect to the line `y == x tan theta` :

Let `P (alpha, beta)` be any point and `Q (x_1 y_ 1 )` be its image about the line

`y = x tan theta (RS)` , then `PQ` `bot` `RS`

` ( `Slope of `PQ ) xx ( `Slope of `RS ) = -1`

`(y_1 - beta)/(x_1-alpha) xx tan theta = - 1`

`=> y_1 -beta=(alpha-x_1 ) cot theta` ....................... `(1)`

and mid point of `PQ` lie on `y = x tan theta`

i.e., `((y_2+beta)/2)=((x_1+alpha)/2)tan theta`

or `y_1 + beta = (x_1 +alpha) tan theta` .......................... `(2)`

Solving (1) and (2), we get

`x_1 = a cos 2 theta + beta sin 2 theta`

`y_1 = alpha sin 2 theta - beta cos 2 theta`

`Q =(alpha cos 2 theta + beta sin 2 theta, alpha sin 2 theta - beta cos 2 theta)`