Common Tangents — Internal, external
Circles
8
JEE Qs
8%
Hard
75
min
Master the geometric interpretation of common tangents and their conditions as it simplifies problem-solving significantly.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always draw a diagram to visualize the relative positions of the circles and the type of common tangents involved.
- 2The intersection points of direct and transverse common tangents always lie on the line joining the centers of the circles.
- 3The lengths of direct and transverse common tangents can be derived using Pythagoras theorem by forming a right-angled trapezoid with centers and tangent points.
- 4To find the actual equations of common tangents, use the general form of a line passing through the intersection point of common tangents, then apply the condition that the perpendicular distance from the center to the tangent equals the radius (p=r).
- 5Remember to consider edge cases such as circles touching externally/internally, concentric circles, or one circle completely inside another.
⚠️ Common Mistakes
- ✕Confusing direct (external) and transverse (internal) common tangents, especially their length formulas and intersection point ratios.
- ✕Incorrectly identifying the number of common tangents for different relative positions of circles.
- ✕Making algebraic errors when calculating the coordinates of the centers, radii, or the distance between centers.
- ✕Forgetting to check the condition for the existence of common tangents before proceeding to calculate their equations or lengths.
- ✕Sign errors in the section formula for external vs. internal division.
📝 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 10: Straight Lines
- Class 11 Maths Ch 11: Conic Sections (Circles)