Family of Circles — Coaxial, radical axis
Circles
8
JEE Qs
8%
Hard
90
min
Master the geometric interpretations of the radical axis and coaxial systems; many problems can be simplified by visualizing their properties rather than purely algebraic manipulation.
🧮 Key Formulas
✅ Key Points for JEE
- 1The radical axis of two circles is the locus of points from which tangents drawn to both circles are of equal length.
- 2The radical axis is always a straight line and is perpendicular to the line joining the centers of the two circles.
- 3For a coaxial system of circles, every pair of circles in the system shares the same radical axis.
- 4The radical center of three circles (no two concentric) is the point where the three radical axes (one for each pair of circles) concur.
- 5A circle that cuts two circles orthogonally will have its center on their radical axis. If it cuts three circles orthogonally, its center will be the radical center of the three circles.
⚠️ Common Mistakes
- ✕Confusing the equation of the radical axis (S1 - S2 = 0) with the family of circles passing through intersection points (S1 + λS2 = 0).
- ✕Forgetting to ensure the coefficients of x^2 and y^2 are 1 before calculating the radical axis (S1-S2=0).
- ✕Algebraic errors when determining the specific circle from a family using given conditions (e.g., passing through a point, tangent to a line).
- ✕Incorrectly identifying the type of family of circles required for a given problem statement (e.g., through two points vs. general coaxial system).
📝 Practice Questions
See allQ1. Let circle C be the image of x2 + y2 −2x + 4y −4 = 0 in the line 2x −3y + 5 = 0 and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C . If B(α, β), with β < 4, lies on C such that the length of the are AB is (1/6)th of the perimeter of C , then β −√3α is equal to (1) 3 + √3 (2) 4 (3) 4 −√3 (4) 3
Q19.Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D , then the area of the quadrilateral ADBC is equal to : (1) √14 (2) 3√7 (3) 2√14 (4) 5√7
Q18.A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (α, β), then 3β −2α is equal to : (1) 10 (2) 15 (3) 12 (4) 14
Q21.Let the circle C touch the line x −y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length 4 along the line −3x + 2y = 1. Let H be the hyperbola x2 −y2 = 1, whose one of the foci is the √13 α2 β2 centre of C and the length of the transverse axis is the diameter of C . Then 2α2 + 3β2 is equal to ______
Q6. Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 −αx + βy + γ = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to (1) (γ, β2 −4α) (2) (α, β2 + 4γ) (3) (γ, β2 + 4α) (4) (α, β2 −4γ) 2x
Q15.Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C , whose mid-point is (1, 2), is : (1) √3 (2) 2√2 (3) 2√3 (4) 4√2
NCERT Chapters
- Class 11 Maths Ch 11: Conic Sections