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Question
Find the equation of the line which satisfy the given condition:
Passing though `(2, 2sqrt3)` and is inclined with the x-axis at an angle of 75°.
Solution
Since, the line is inclined at 75° to the x-axis, then the slope of the line is
m = tan 75° = tan (45° + 30°)
= `("tan" 45° + "tan" 30°)/(1 - "tan" 45° "tan" 30°)`
= `(1 + 1/sqrt3)/(1 - 1. 1/sqrt3)`
= `((sqrt3 + 1)/sqrt3)/((sqrt3 - 1)/sqrt3)`
= `(sqrt3 + 1)/(sqrt3 - 1)`
We know that the equation of the line m passing through the point (x0, y0) whose slope is (y − y0) = m(x − x0).
If a line passes through an angle of 75° with the x-axis and the equation of the line is given as.
`("y" - 2sqrt3) = (sqrt3 + 1)/(sqrt3 -1) ("x"- 2)`
`("y" - 2sqrt3) (sqrt3 - 1) = (sqrt3 + 1) ("x" - 2)`
`"y"(sqrt3 - 1) - 2sqrt3(sqrt3 - 1) = "x"(sqrt3 + 1) - 2(sqrt3 + 1)`
`(sqrt3 + 1) "x" - (sqrt3 - 1) "y" = 2sqrt3 + 2 - 6 + 2sqrt3`
`(sqrt3 + 1) "x" - (sqrt3 - 1) "y" = 4sqrt3 - 4`
∴ `(sqrt3 + 1) "x" - (sqrt3 - 1) "y" = 4(sqrt3 - 1)`
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