Find the condition for the chord lx+my=1 of the circle x^2+y^2=a^2 to subtend a right angle at the origin
Answers
Given : chord lx+my=1 of the circle x^2+y^2=a^2 to subtend a right angle at the origin
To Find : condition for the chord lx+my=1
Solution:
x² + y² = a²
center (0, 0) & radius = a
x² + y² = a²
=> x² + y² = a² .1²
lx+my=1
=> x² + y² = a²(lx + my)²
=> x² + y² = a² (l²x² + m²y² + 2lmxy)
=> x² + y² = a²l²x² + a²m²y² + 2a²lmxy
=> a²l²x² + a²m²y² + 2a²lmxy - x² - y² = 0
=> (a²l² - 1)x² + 2a²lmxy + (a²m² - 1)y² = 0
lx+my=1 subtends right angle at origin
Hence coefficient of x² / coefficient of y² = - 1
=> (a²l² - 1) / (a²m² - 1) = - 1
=> (a²l² - 1) = - (a²m² - 1)
=> a²l² - 1 + a²m² - 1 = 0
=> a²l² + a²m² = 2
=> a²(l² + m²) = 2
a²(l² + m²) = 2 is the required condition
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