CUET UG Mathematics Coordinate Geometry Circles Equation Tangent Normal Chord of Contact

Must‑Know (15–20 detailed bullets)

• General equation of a circle: ( x^2 + y^2 + 2gx + 2fy + c = 0 ), with center ( (-g, -f) ) and radius ( \sqrt{g^2 + f^2 - c} ). Example: ( x^2 + y^2 - 4x + 6y - 12 = 0 ) has center (2, –3) and radius 5.

• Standard equation of a circle with center (h, k) and radius r: ( (x - h)^2 + (y - k)^2 = r^2 ). Example: Circle with center (3, –2) and radius 4: ( (x - 3)^2 + (y + 2)^2 = 16 ).

• If a circle passes through the origin, then in general form, ( c = 0 ). Example: ( x^2 + y^2 - 6x + 8y = 0 ) passes through (0, 0).

• Equation of a circle on a diameter with endpoints ( (x_1, y_1) ) and ( (x_2, y_2) ): ( (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 ). Example: Endpoints (1, 2) and (3, 4): ( (x - 1)(x - 3) + (y - 2)(y - 4) = 0 ).

• Condition for a second-degree equation ( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 ) to represent a circle: ( a = b \neq 0 ) and ( h = 0 ). Example: ( 3x^2 + 3y^2 + 4x - 6y + 2 = 0 ) is a circle.

• Length of tangent from an external point ( (x_1, y_1) ) to the circle ( x^2 + y^2 + 2gx + 2fy + c = 0 ) is ( \sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c} ). Example: From (2, 3) to ( x^2 + y^2 - 4x + 6y - 12 = 0 ), length is ( \sqrt{4 + 9 - 8 + 18 - 12} = \sqrt{11} ).

• Equation of tangent to circle ( x^2 + y^2 = r^2 ) at point ( (x_1, y_1) ) on the circle: ( xx_1 + yy_1 = r^2 ). Example: Tangent at (3, 4) to ( x^2 + y^2 = 25 ) is ( 3x + 4y = 25 ).

• Equation of tangent to circle ( (x - h)^2 + (y - k)^2 = r^2 ) at ( (x_1, y_1) ): ( (x - h)(x_1 - h) + (y - k)(y_1 - k) = r^2 ). Example: At (5, 1) on ( (x - 3)^2 + (y + 1)^2 = 8 ): ( (x - 3)(2) + (y + 1)(2) = 8 ).

• Slope form of tangent to ( x^2 + y^2 = r^2 ): ( y = mx \pm r\sqrt{1 + m^2} ). Example: Tangents with slope 1 to ( x^2 + y^2 = 9 ): ( y = x \pm 3\sqrt{2} ).

• Condition for line ( y = mx + c ) to be tangent to ( x^2 + y^2 = r^2 ): ( c = \pm r\sqrt{1 + m^2} ). Example: ( y = 2x + c ) touches ( x^2 + y^2 = 5 ) if ( c = \pm \sqrt{5} \cdot \sqrt{5} = \pm 5 ).

• Equation of normal to a circle at a point on the circle passes through the center. Example: Normal at (1, 1) on ( x^2 + y^2 = 2 ) is line through (0, 0) and (1, 1): ( y = x ).

• The normal to the circle ( x^2 + y^2 + 2gx + 2fy + c = 0 ) at ( (x_1, y_1) ) has direction ratios ( (x_1 + g, y_1 + f) ), same as radius vector.

• Length of chord intercepted by line ( y = mx + c ) on circle ( x^2 + y^2 = r^2 ): ( 2\sqrt{r^2 - d^2} ), where ( d ) is perpendicular distance from center to line. Example: Line ( y = x ) cuts ( x^2 + y^2 = 16 ); ( d = 0 ), so chord length = 8.

• Equation of chord of contact from external point ( (x_1, y_1) ) to circle ( x^2 + y^2 = r^2 ): ( xx_1 + yy_1 = r^2 ). Example: From (4, 5) to ( x^2 + y^2 = 25 ): ( 4x + 5y = 25 ).

• Chord of contact from ( (x_1, y_1) ) to circle ( x^2 + y^2 + 2gx + 2fy + c = 0 ) is ( xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 ). Example: From (2, 3) to ( x^2 + y^2 - 4x + 6y - 12 = 0 ): ( 2x + 3y -2(x + 2) + 3(y + 3) - 12 = 0 ) → simplify.

• Two tangents can be drawn from any external point to a circle; they are equal in length. Example: From (5, 0) to ( x^2 + y^2 = 9 ), both tangents have length ( \sqrt{25 + 0 - 9} = 4 ).

• Angle between tangents from ( (x_1, y_1) ) to ( x^2 + y^2 = r^2 ): ( 2 \tan^{-1}\left( \frac{r}{\sqrt{x_1^2 + y_1^2 - r^2}} \right) ). Example: From (5, 0) to ( x^2 + y^2 = 9 ), angle = ( 2 \tan^{-1}(3/4) ).

• Director circle of ( x^2 + y^2 = r^2 ) is ( x^2 + y^2 = 2r^2 ); it is the locus of points from which perpendicular tangents are drawn.

• If two circles touch externally, distance between centers = sum of radii. Example: ( (x - 1)^2 + (y - 2)^2 = 4 ) and ( (x - 4)^2 + (y - 2)^2 = 1 ): distance = 3, sum = 2 + 1 = 3 → touch externally.

• If two circles touch internally, distance between centers = difference of radii. Example: ( (x - 1)^2 + (y - 1)^2 = 9 ) and ( (x - 1)^2 + (y - 1)^2 = 1 ): distance = 0, difference = 8 → not valid; correct example: centers (0,0) r=5, (3,0) r=2 → distance=3, difference=3 → touch internally.

Difficulty Level

Intermediate — Requires understanding of equations, geometric interpretation, and algebraic manipulation; direct formula application and word problems are common.

Common CUET Traps

• Trap: Assuming ( x^2 + y^2 + 2gx + 2fy + c = 0 ) always represents a real circle.
  Avoid: Check ( g^2 + f^2 - c > 0 ) for real circle; if =0, point circle; if <0, no real locus.

• Trap: Using ( xx_1 + yy_1 = r^2 ) for tangent when the circle is not centered at origin.
  Avoid: Use ( (x - h)(x_1 - h) + (y - k)(y_1 - k) = r^2 ) for center (h,k); shift origin if needed.

• Trap: Confusing chord of contact with polar or diameter.
  Avoid: Chord of contact is only defined from an external point and gives the line joining points of tangency; verify point is outside the circle.

Practice MCQs (5 questions)

Q1. The equation ( x^2 + y^2 - 6x + 8y + 25 = 0 ) represents:
A) A circle of radius 1
B) A point
C) A circle of radius 0
D) No real locus

Answer: B
Explanation: ( g = -3, f = 4, c = 25 ); ( g^2 + f^2 - c = 9 + 16 - 25 = 0 ) → point circle.
Why others fail: Option D is tempting if student miscalculates discriminant as negative.

Q2. What is the length of the tangent from (4, 3) to the circle ( x^2 + y^2 = 16 )?
A) 3
B) 4
C) 5
D) 6

Answer: A
Explanation: Length = ( \sqrt{4^2 + 3^2 - 16} = \sqrt{16 + 9 - 16} = \sqrt{9} = 3 ).
Why others fail: Option C (5) is tempting as it's the distance from origin to (4,3), but not the tangent length.

Q3. The equation of the tangent to the circle ( x^2 + y^2 = 25 ) at (–3, 4) is:
A) ( 3x - 4y = 25 )
B) ( -3x + 4y = 25 )
C) ( 3x + 4y = 7 )
D) ( -3x + 4y = 7 )

Answer: B
Explanation: Tangent: ( x(-3) + y(4) = 25 ) → ( -3x + 4y = 25 ).
Why others fail: Option A has sign error in coefficient; students often forget sign of point coordinates.

Q4. The chord of contact from (2, –3) to the circle ( x^2 + y^2 = 4 ) is:
A) ( 2x - 3y = 4 )
B) ( 2x + 3y = 4 )
C) ( x - y = 1 )
D) ( 3x - 2y = 4 )

Answer: A
Explanation: Chord of contact: ( x(2) + y(-3) = 4 ) → ( 2x - 3y = 4 ).
Why others fail: Option B is tempting if sign of y-coordinate is ignored.

Q5. The angle between the tangents drawn from (5, 0) to the circle ( x^2 + y^2 = 9 ) is:
A) ( 2 \sin^{-1}(3/5) )
B) ( 2 \tan^{-1}(3/4) )
C) ( 2 \cos^{-1}(3/5) )
D) ( 2 \tan^{-1}(4/3) )

Answer: B
Explanation: Tangent length = 4, radius = 3; angle = ( 2 \tan^{-1}(r / \text{length}) = 2 \tan^{-1}(3/4) ).
Why others fail: Option A uses sine instead of tangent; common confusion in right-triangle ratios.

Last‑Minute Revision (15–20 one‑liners)

• ⚠️ General circle equation: ( x^2 + y^2 + 2gx + 2fy + c = 0 ); center ( (-g, -f) ), radius ( \sqrt{g^2 + f^2 - c} ).
• ⚠️ Circle through origin ⇒ constant term ( c = 0 ) in general form.
• ⚠️ Diameter form: ( (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 ).
• ⚠️ For ( ax^2 + 2hxy + by^2 + \cdots = 0 ) to be circle: ( a = b ), ( h = 0 ).
• ⚠️ Tangent at ( (x_1, y_1) ) on ( x^2 + y^2 = r^2 ): ( xx_1 + yy_1 = r^2 ).
• ⚠️ Slope form tangent: ( y = mx \pm r\sqrt{1 + m^2} ) for circle at origin.
• ⚠️ Condition for tangency: ( c = \pm r\sqrt{1 + m^2} ) for line ( y = mx + c ).
• ⚠️ Normal to circle at a point passes through the center — always.
• ⚠️ Length of tangent from ( (x_1, y_1) ): ( \sqrt{S_1} ), where ( S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c ).
• ⚠️ Chord of contact from ( (x_1, y_1) ): ( T = 0 ), i.e., ( xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 ).
• ⚠️ For ( x^2 + y^2 = r^2 ), chord of contact: ( xx_1 + yy_1 = r^2 ).
• ⚠️ Two equal tangents from external point — length formula is symmetric.
• ⚠️ Angle between tangents: ( 2 \tan^{-1}(r / \sqrt{S_1}) ).
• ⚠️ Director circle of ( x^2 + y^2 = r^2 ) is ( x^2 + y^2 = 2r^2 ).
• ⚠️ External touch: distance = ( r_1 + r_2 ); internal touch: distance = ( |r_1 - r_2| ).
• ⚠️ If ( g^2 + f^2 - c = 0 ), circle is a point; if <0, no real graph.
• ⚠️ Verify from NCERT: exact derivation of tangent using discriminant method.
• ⚠️ Chord length = ( 2\sqrt{r^2 - d^2} ), d = perpendicular from center to chord.
• ⚠️ Mnemonic: "Tangent touches, Normal goes to center" — helps recall geometric behavior.
• ⚠️ For MCQs: always check if point is outside, on, or inside circle before applying tangent formulas.