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MathsMediumClass 12

Tangent from External Point — Length, condition

Circles

8

JEE Qs

8%

Hard

40

min

Master the direct application of S1 for length and point position, as it's fundamental for many advanced circle problems.

🧮 Key Formulas

For a circle S = x^2 + y^2 + 2gx + 2fy + c = 0, the power of a point P(x1, y1) with respect to the circle is S1 = x1^2 + y1^2 + 2gx1 + 2fy1 + c.
The length of the tangent from an external point P(x1, y1) to the circle S=0 is L = sqrt(S1).
Condition for point P(x1, y1) relative to circle S=0: - If S1 > 0, P is outside the circle (two real tangents). - If S1 = 0, P is on the circle (one tangent, the line itself). - If S1 < 0, P is inside the circle (no real tangents).
Equation of the pair of tangents from an external point P(x1, y1) to the circle S=0 is T^2 = S S1, where T = xx1 + yy1 + g(x+x1) + f(y+y1) + c.

✅ Key Points for JEE

  • 1Always ensure the circle's equation is in standard form (coefficients of x^2 and y^2 are unity) before calculating S1.
  • 2The expression S1 represents the 'power of the point' with respect to the circle; its square root gives the length of the tangent.
  • 3Tangents drawn from an external point to a circle are equal in length and subtend equal angles at the center.
  • 4The equation T^2 = S S1 is a powerful tool to directly find the combined equation of both tangents from an external point.

⚠️ Common Mistakes

  • ✕Forgetting to take the square root of S1 when asked for the *length* of the tangent.
  • ✕Failing to normalize the circle equation (making coefficients of x^2 and y^2 unity) before substituting into S1.
  • ✕Incorrectly interpreting the sign of S1 for the position of the point (S1 > 0 for outside, not inside).

📝 Practice Questions

See all

Q1. 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

2025·MCQHard

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

2025·MCQMedium

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

2025·MCQHard

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 ______

2025·NumericalHard

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

2025·MCQMedium

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

2025·MCQMedium

NCERT Chapters

  • Class 11 Maths Ch 11: Conic Sections (Circles)