Pair of Tangents — Chord of contact
Circles
8
JEE Qs
8%
Hard
60
min
Master the general forms T=0 and SS1=T^2 for any conic, and understand their geometric interpretation to solve complex problems efficiently, avoiding algebraic errors.
🧮 Key Formulas
✅ Key Points for JEE
- 1The equation T=0 represents the chord of contact when the point P(x1, y1) is external to the circle. It's also the polar of P(x1, y1) with respect to the circle.
- 2The equation SS1=T^2 is a second-degree equation that represents the two straight lines (tangents) from the external point P(x1, y1) to the circle.
- 3For the chord of contact to exist as a physical chord, the point P(x1, y1) must be external to the circle (S1 > 0). If S1=0, P is on the circle, and the chord of contact becomes the tangent at P. If S1<0, P is inside, and there are no real tangents, so no real chord of contact.
- 4The T=0 and SS1=T^2 forms are general and apply to all conic sections (parabola, ellipse, hyperbola) with their respective S and T forms.
- 5Problems often combine concepts like finding the locus of the point P, the area of the quadrilateral formed by tangents and radii, or the properties of the triangle formed by the tangents and the chord of contact.
⚠️ Common Mistakes
- ✕Confusing the point (x1, y1) being on the circle versus being an external point when applying T=0. T=0 is the tangent equation if (x1,y1) is on the circle, but the chord of contact equation if (x1,y1) is external.
- ✕Incorrectly expanding or simplifying the SS1=T^2 equation, which is algebraically intensive. Students often make sign errors or miscalculate coefficients.
- ✕Applying the formulas without ensuring the given circle equation is in the standard form (e.g., coefficient of x^2 and y^2 is 1).
- ✕Not understanding the geometric meaning of T=0 (a line) and SS1=T^2 (a pair of lines) and S1 (a constant representing square of length of tangent).
📝 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)