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प्रश्न
Find the measure of the acute angle between the line represented by:
4x2 + 5xy + y2 = 0
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उत्तर
Comparing the equation
4x2 + 5xy + y2 = 0 with
ax2 + 2hxy + by2 = 0, we get,
a = 4, 2h = 5 i.e. h = `5/2` and b = 1
Let θ be the acute angle between the lines.
∴ tan θ = `|(2 sqrt("h"^2 - "ab"))/("a + b")|`
= `|(2 sqrt((5/2)^2 - 4(1)))/(4 + 1)|`
= `|(2 sqrt((25/4) - 4))/(5)|`
= `|(2 xx 3/2)/(5)|`
∴ tan θ = `3/5`
∴ θ = tan-1 `(3/5)`
Notes
This can be written as `2x^2 – 7xy + 3y^2 = (2x – y)(x-3y)`
Therefore, the given lines are (2x – y) = 0 and (x-3y) = 0.
Now, here, a = 2, h = `7/2` and b = 3.
Hence, the angle between the lines is given by tan θ = `2sqrt{(7/2)2 – 6}/5 = 1.`
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