The ends A , B of a fixed straight line of length ‘a’ and ends A ′ and B ′ of another fixed straight line of length ‘b’ slide upon the axis of X & the axis of Y (one end on axis of X & the other on axis of Y). Find the locus of the centre of the circle passing through A, B, A ′ and B ′ .
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(2ax − 2by)² + (2bx − 2ay)² = (a² − b²)²
Let ∠ OA ′ B ′ = φ and ∠ OAB = θ
⇒ θ + φ = 
and ∠ OBA = φ
length of AB is ‘a’ and length of A ′ B ′ is ‘b’
∴ from the figure
A ′ (b cos φ , 0) and A(a cos θ , 0)
Similarly B(0, a sin θ ) and B ′ (0, b sin φ )
Let c(h, k) be the centre of circle
∴ 2h = a cos θ + b cos φ
φ = 
– θ
∴ 2h = a cos θ + b sin θ ........(i)
and 2k = a sin θ + b sin φ
φ = 
– θ
∴ 2k = a sin θ + b cos θ ........(ii)
on solving (i) and (ii), we get cos θ = 
and sin θ = 
sin 2 θ + cos 2 θ = 1
∴ locus of C(h, k) is (2ax − 2by)² + (2bx − 2ay)² = (a² − b²)²
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems