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Question

Find the tangent of the angle between the lines whose intercepts on the axes are respectively a,-b and b, -a


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Solution

Step 1: Finding equation of line that intercepts x-axis and y-axis at a and -b respectively

Equation of a line is given as,

y=mx+c

where m and c are the slope and y-intercept of the line.

Given, line intercepts x-axis at a, that means, y=0 when x=a

0=m1·a+c1m1=-c1a

where m1 and c1 are the slope and y-intercept of the line.

Also given, line intercepts y-axis at -b, that means, y=a when x=0

-b=m1·0+c1c1=-b

But,

m1=-c1am1=ba

Step 2: Finding equation of line that intercepts x-axis and y-axis at b and -a respectively

Given, line intercepts x-axis at b, that means, y=0 when x=b

0=m2·b+c2m2=-c2b

where m2 and c2 are the slope and y-intercept of the line.

Also given, line intercepts y-axis at -a, that means, y=b when x=0

-a=m2·0+c2c2=-a

But,

m2=-c2am2=ab

Step 3: Finding the tangent of the angle between the lines

Let θ be the angle between the two lines.

Tangent of angle between the two lines is given as,

tanθ=m2-m11+m2m1tan(θ)=ab-ba1+ab·batan(θ)=a2-b2ab2tan(θ)=a2-b22ab

Therefore, the tangent of the angle between the lines, tan(θ)=a2-b22ab


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