Question

The graph of $y - x$ against $y + x$ is as shown (straight line with slope>0 through origin). Which of the given five graphs shows $y$ against $x$?

• A.

  

• B.

  

• C.

  

• D.

  

• E.

  

Hint:

Convert given transformed axes equations back to $x$-$y$ form to identify the slope and intercept.

Answer 1

The Correct Option is D

Let $u = y - x$ and $v = y + x$. The given graph is $u = m v$ with $m>0$. Substituting: \[ y - x = m (y + x) \] \[ y - x = my + mx \] \[ y - my = x + mx \] \[ y(1 - m) = x(1 + m) \] \[ y = \frac{1 + m}{1 - m} x \] For $m>0$, if $m>1$, slope $\frac{1+m}{1-m}$ is negative, giving a downward sloping line through origin. The drawn graph slope>0 in $u-v$ space and intercepts at origin means $m>1$ is valid. \[ \boxed{\text{Graph (4)}} \]