A line with the equation y = px + q is reflected over the line y = x. Is the reflection of this line parallel to the line y = mx + n?
(1) m = p + 2
(2) m = 3p
(1) m = p + 2
(2) m = 3p
If a line is reflected across the line y=x, its slope is the reciprocal of the original slope. So, for y=mx+n to be parallel to y=px+q, it must be true that m=1/p (as long as p is not zero).
Statement 1: m=p+2. It is easy to pick values of p that make m=1/p false. For example, p=2 gives m=4, and 4 is not equal to 1/2. So we have to determine if there are ANY cases where m=p+2 AND m=1/p. Set the equations equal to each other:
p+2=1/p
p^2+2p=1
p^2+2p-1=0
the discriminant of this equation is b^2-4ac=8. That is positive, so the equation does have real solutions. Thus there are values of p such that m=p+2 and m=1/p. INSUFFICIENT.
Statement 2: m=3p. once again, it is easy to find values of p that don't work. p=2 makes m=6, and 6 is not equal to 1/2. Solving m=3p and m=1/p simultaneously gives us
3p=1/p
3p^2=1
p^2=1/3
p=+-√3/3
So, INSUFFICIENT.
Statements 1 & 2 together: 3p=p+2, 2p=2, p=1. m=3. Does m=1/p? 3 does not equal 1, so no. So they are definitely not parallel. SUFFICIENT.
Ans: C
