Position of a Point w.r.t a Circle

The point P(x1 , y1) lies outside , on , or inside a circle

S ≡ x2 + y2 + 2gx + 2fy + c = 0 , according as

S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c > = or < 0.

Length of the intercept made by a circle on a line:

If the line l meets the circle S with centre C and radius ‘ a ‘ in two distinct points A and B and if d is the perpendicular distance of C from the line l .

The length of the intercept made by the circle on the line = |AB| = 2 √(a2 − d2)

To find the point of intersection of a line y = mx + c with a circle x2 + y2 = a2 we need to solve both the curves

i.e. roots of equation x2 + (mx + c)2 = a2 gives x coordinates of the point of intersection.

Now following cases arise:

(i) Discriminant > 0 => two distinct and real points of intersection.

(ii) Discriminant = 0 => coincident roots i.e. line is tangent to the circle.

(iii) Discriminant < 0 => no real point of intersection.

Also Read :

Equations of Circle in various forms
Parametric Equation of a Circle
Equation of Tangent & Normal on the Circle
Chord of Contact
Radical Axis of two circles
Equation of Family of Circles
External & Internal Contacts of Circles

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