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Question

Show that the tangent of an angle between the lines xa+yb=1 and xayb=1 is 2aba2b2


Solution

Let the line xa+yb=1 be AB and the line xayb=1 be CD.

Equation of AB, bx+ayab=1

ay=bx+ab

y=bxa+b

Therefore, m1=ba

Similarly, the equation of CD, bxayab=1

bxay=ab

ay=bxaa

Therefore, m2=ba

The tangent of angle between the lines AB and CD is

tan θ=m1m21+m1m2=∣ ∣baba1+(ba)(ba)∣ ∣

=2b9a2b2a=2aba2b2

The tangent of the angle between the lines =2aba2b2

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