My Math Forum [ASK] Reflection of a Line by Another Line

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 March 31st, 2018, 06:29 AM #1 Senior Member     Joined: Nov 2010 From: Indonesia Posts: 2,001 Thanks: 132 Math Focus: Trigonometry [ASK] Reflection of a Line by Another Line The reflection of the line 5x - 7y - 13 = 0 by the line y = -x is .... A. 7x + 5y - 13 = 0 B. 7x + 5y + 13 = 0 C. 7x - 5y - 13 = 0 D. 7x - 5y + 13 = 0 E. 7y + 5x + 13 = 0 This one I totally have no idea. Like, at all.
 March 31st, 2018, 11:36 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 No idea? Do you know what a line is? Do you know what symmetric means? That's quite a lot to be getting on with! A line reflects into a line. And two points determine a line. Let $\displaystyle (0, y_0)$ be any point on the y-axis. Then "reflection in the line y- x" maps it to the point $\displaystyle (-y_0, 0)$. And any point $\displaystyle (x_0, 0)$ on the x-axis is mapped to the point $\displaystyle (0, -x_0)$. Let x= 0 in 5x- 7y- 13= 0. then -7y- 13= 0 so y= -13/7. The point (0, -13/7) is on the given line and reflection in the line y= -x maps it to (13/7, 0). Let y= 0 in 5x- 7y- 13= 0. Then 5x- 13= 0 so x= 13/5. The point (13/5, 0) is o the given line and reflection in the line y= -x maps it to (0, -13/5). So the question is just to find the equation of the line through (13/7, 0) and (0, -13/5). Any non-vertical line can be written in the form y= mx+ b. Taking x= 13/7, y= 0, we have 13m/7+ b= 0. Taking x= 0, y= -13/5, m(0)+b= b= -13/5. Then 13m/7- 13/5= 0 so 13m/7= 13/5. m/7= 1/5 so m= 7/5. The equation of the line is y= (7/5)x- 13/5. 5y= 7x- 13 so 7x- 5y= 13.
March 31st, 2018, 05:08 PM   #3
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Quote:
 Originally Posted by Country Boy No idea? Do you know what a line is? Do you know what symmetric means? That's quite a lot to be getting on with!
I know, in fact I can solve that graphically by manually find the reflection's equation, but I didn't know how to do it algebraically. Anyway, thanks for your help.

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