the length of the chord formed by x^2+y^2=a^2 on the line x cos alpha + y sin alpha = p is
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Answer:
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Given : chord formed by x^2+y^2=a^2 on the line x cos alpha + y sin alpha = p
To Find : Length of chord
Solution:
circle x² + y² = a²
center O = ( 0 , 0)
Radius = a
A and B are the end points of chord
Then M is the mid point of chord
AM = BM = AB/2
AM² + OM² = OA² or BM² + OM² = OB²
OA = OB = Radius = a
OM ⊥ AB
Hence perpendicular distance of O(0,0) from x cosα + y sinα = p
x cosα + y sinα - p = 0 point (0, 0)
= | (0 * cosα + 0 * sinα - p ) / (cos²α + sin²α) |
cos²α + sin²α = 1
= | - p / 1|
= | - p|
OM = | - p|
OM² = p²
AM² + p² = a²
=> AM² = a² - p²
=> AM = √a² - p²
2AM = 2√a² - p²
=> AB = 2√a² - p²
Hence length of the chord formed = 2√a² - p²
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