Radical Axis & Radical Centre
Circles
8
JEE Qs
8%
Hard
60
min
Master the geometric interpretation of radical axis as the locus of equal power, as it simplifies complex problems involving tangents and multiple circles.
🧮 Key Formulas
✅ Key Points for JEE
- 1The radical axis is the locus of all points from which the lengths of tangents drawn to two circles are equal (or equivalently, the power of the point with respect to both circles is equal).
- 2Before finding the radical axis S1 - S2 = 0, always ensure the coefficients of x^2 and y^2 in both circle equations (S1 and S2) are unity. If not, divide the entire equation by that coefficient first.
- 3The radical axis is always perpendicular to the line joining the centres of the two circles.
- 4If two circles intersect, their radical axis is their common chord. If they touch externally or internally, their radical axis is their common tangent at the point of contact. If they are disjoint, the radical axis does not intersect either circle.
- 5The radical centre of three non-coaxial circles is the point of intersection of the three radical axes formed by taking the circles pairwise. This point has equal power with respect to all three circles.
⚠️ Common Mistakes
- ✕Not standardizing the circle equations (i.e., making coefficients of x^2 and y^2 equal to 1) before subtracting to find the radical axis equation.
- ✕Confusing the equation of the radical axis (S1 - S2 = 0) with the family of circles passing through the intersection of two circles (S1 + λS2 = 0).
- ✕Misinterpreting the geometric significance of the radical axis, particularly its relation to tangents and power of a point.
- ✕Incorrectly finding the radical centre by not using pairwise radical axes or miscalculating the intersection point.
📝 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