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qudoniselmore0
20.01.2020 •
Mathematics
Acircle has equation
, where k > 0.
the straight line l with the equation
cuts the circle at two distinct points.
prove that ![k\ \textgreater \ \frac{a^{2} }{1+a^{2} }](/tex.php?f=k\ \textgreater \ \frac{a^{2} }{1+a^{2} })
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Ответ:
Step-by-step explanation:
(x − 1)² + y² = k
y = ax
When they intersect:
(x − 1)² + (ax)² = k
x² − 2x + 1 + a²x² = k
(1 + a²) x² − 2x + 1 − k = 0
For this to have two real solutions, the discriminant must be positive.
b² − 4ac > 0
(-2)² − 4 (1 + a²) (1 − k) > 0
4 + 4 (1 + a²) (k − 1) > 0
4 (1 + a²) (k − 1) > -4
(1 + a²) (k − 1) > -1
k − 1 > -1 / (1 + a²)
k > 1 − 1 / (1 + a²)
k > (1 + a² − 1) / (1 + a²)
k > a² / (1 + a²)
Ответ: