Q.Find the angle between the circles x2+y2−4x−6y−3=0 and x2+y2+2x+2y−2=0.
Telangana TsbieTextbookSubjectiveImportance★★★★★
3% · 1/37 Questions
✓ Free question
Concept understanding — Angle Between Two Intersecting Circles
The angle between two intersecting circles at a common point P is defined as the angle between their tangents at P -- equivalently (since each tangent is perpendicular to its own radius), the angle between the two radii C1P and C2P, where C1,C2 are the centres.
Let the circles have centres C1,C2, radii r1,r2, and let d=C1C2 be the distance between the centres. In triangle C1PC2, the sides are C1P=r1, C2P=r2, and C1C2=d, and the included angle at P is the angle θ between the circles. The cosine rule in this triangle gives
This can be rewritten purely in terms of the circles' coefficients. For S≡x2+y2+2gx+2fy+c=0 and S′≡x2+y2+2g′x+2f′y+c′=0, the centres are (−g,−f) and (−g′,−f′), so d2=(g−g′)2+(f−f′)2, while r12=g2+f2−c and r22=g′2+f′2−c′. Substituting and expanding:
The two circles' common chord (the line through their two points of intersection, when they meet in two real points) is obtained by subtracting one equation from the other: S−S′=0 is linear in x,y (the x2 and y2 terms cancel), and since every point common to S=0 and S′=0 satisfies S−S′=0 as well, this line passes through both intersection points. When the two circles instead touch (meet at exactly one point), S−S′=0 is their common tangent at the point of contact.
Find each centre and radius, then use cosθ=2r1r2d2−r12−r22.
✓Final answer
θ=cos−1(165).
Step 1. For S:x2+y2−4x−6y−3=0: g=−2,f=−3,c=−3, so centre C1=(2,3) and r12=g2+f2−c=4+9+3=16, i.e. r1=4.
Step 2. For S′:x2+y2+2x+2y−2=0: g′=1,f′=1,c′=−2, so centre C2=(−1,−1) and r22=1+1+2=4, i.e. r2=2.
Step 3. Distance between centres: d2=(2−(−1))2+(3−(−1))2=32+42=9+16=25, so d=5.
Step 4. Apply the angle formula:
cosθ=2r1r2d2−r12−r22=2(4)(2)25−16−4=165.
Step 5 (check via the coefficient form).cosθ=2r1r2c+c′−2gg′−2ff′=16−3−2−2(−2)(1)−2(−3)(1)=16−5+4+6=165. Matches.
✓Final answer
The angle between the circles is θ=cos−1(165).
Angle between two intersecting circles, via the cosine rule in the triangle formed by the two centres and a point of intersection.
Forgetting the sign convention r2=g2+f2−c (subtracting c, not adding it).
Arithmetic slip computing d2 from the two centres.