Circles — Tangent Chord of Contact
Circles
54
JEE Qs
22%
Hard
60
min
Master the versatile T=0 form and its geometric interpretations for both tangents and chords of contact, as it is a central concept that simplifies many problems.
🧮 Key Formulas
✅ Key Points for JEE
- 1The equation T=0 represents both the tangent at (x1, y1) when (x1, y1) is on the circle, AND the chord of contact when (x1, y1) is an external point. Understanding this duality is crucial.
- 2The chord of contact joins the points where tangents from an external point touch the circle. Its equation is derived from the fact that these points lie on both the circle and the tangents.
- 3Do not confuse the chord of contact (from an external point) with a chord whose midpoint is given. The latter uses T=S1, while the former uses T=0.
- 4The concept of 'power of a point' is directly linked to the length of the tangent from an external point to a circle (L = sqrt(S1)).
- 5Problems often involve finding the locus of an external point from which the chord of contact passes through a fixed point, or subtends a certain angle at the center.
⚠️ Common Mistakes
- ✕Interchanging the formulas for chord of contact (T=0) and chord with a given midpoint (T=S1).
- ✕Incorrectly applying the (x1, y1) point; sometimes students treat the point of tangency as (x1, y1) instead of the external point for chord of contact.
- ✕Algebraic errors, especially when dealing with the general equation of a circle (x^2 + y^2 + 2gx + 2fy + c = 0) and simplifying the T=0 or T=S1 forms.
📝 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 (specifically, circles portion and basic tangent concepts)