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प्रश्न
The locus of the point of intersection of lines `sqrt(3)x - y - 4sqrt(3)k` = 0 and `sqrt(3)kx + ky - 4sqrt(3)` = 0 for different value of k is a hyperbola whose eccentricity is 2.
विकल्प
True
False
उत्तर
This statement is True.
Explanation:
The given equations are
`sqrt(3)x - y - 4sqrt(3)k` = 0 ......(i)
`sqrt(3)kx + ky - 4sqrt(3)` = 0 ......(ii)
From equation (i) we get
`4sqrt(3)k = sqrt(3)x - y`
∴ `k = (sqrt(3)x - y)/(4sqrt(3))`
Putting the value of k in equation (ii), we get
`sqrt(3)[(sqrt(3)x - y)/(4sqrt(3))]x + [(sqrt(3)x - y)/(4sqrt(3))]y - 4sqrt(3)` = 0
⇒ `((sqrt(3)x - y)/4)x + ((sqrt(3)x - y)/(4sqrt(3)))y - 4sqrt(3)` = 0
⇒ `((3x - sqrt(3)y)x + (sqrt(3)x - y)y - 48)/(4sqrt(3))` = 0
⇒ `3x^2 - sqrt(3)xy + sqrt(3)xy - y^2 - 48` = 0
⇒ `3x^2 - y^2` = 48
⇒ `x^2/16 - y^2/48` = 1 which is a hyperbola.
Here a2 = 16, b2 = 48
We know that b2 = a2(e2 – 1)
⇒ 48 = 16(e2 – 1)
⇒ 3 = e2 – 1
⇒ e2 = 4
⇒ e = 2
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