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The locus of the point of intersection of x/a - y/b = m ; x/a + y/b = 1/m
x/a - y/b = m .............1 x/a + y/b = 1/m .................2 adding both equations 2x/a = m + 1/m ..........................3 subtracting both equations 2y/b = 1/m - m .................4 now , (eq3 )2 - (eq4)2 (2x/a)2 - (2y/b)2 = (m+1/m)2 - (1/m-m)2 = m2 + 1/m2 + 2 - (m2 + 1/m2 - 2) (2x/a)2 - (2y/b)2 = 4 x2/a2 - y2/b2 = 1 this is the equation of hyperbola
x/a - y/b = m .............1
x/a + y/b = 1/m .................2
adding both equations
2x/a = m + 1/m ..........................3
subtracting both equations
2y/b = 1/m - m .................4
now , (eq3 )2 - (eq4)2
(2x/a)2 - (2y/b)2 = (m+1/m)2 - (1/m-m)2
= m2 + 1/m2 + 2 - (m2 + 1/m2 - 2)
(2x/a)2 - (2y/b)2 = 4
x2/a2 - y2/b2 = 1
this is the equation of hyperbola
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