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Normal to a Circle at a Given Point

The normal of a circle at any point is a straight line which is perpendicular to the tangent at the point of contact.
The normal of the circle always passes through the center of the circle.
To find the equation of normal to the circle, x2 + y2 + 2gx + 2fy + c = 0 at the point (x1, y1) on it.
Since normal passes through the center we have slope of normal CP = 68458.png. Hence equation of normal is
yy1 = 68452.png(xx1)
or 68445.png

Chord of contact

From any external point A (x1, y1) draw pair of tangents AP and AQ touching the circle at P(x, y) and Q(x, y) respectively. The line joining P and Q is called chord of contact and its equation is given by xx1 + yy1 = a2 or T = 0, where T = xx1 + yy1a2.
Equation of the chord bisected at a given point Let any chord AB of the circle x2 + y2 + 2gx + 2fy + c = 0 be bisected at D (x1, y1).
Then its equation is xx1 + yy1 + g(x + x1) + f(y + y1) + c = x12 + y12 + 2gx1 + 2fy1 + c or T = S1.
Note: The chord bisected at point (x1y1) is the chord of minimum length passing through the point (x1y1) and at maximum distance from the center.

Pair of tangents

Let the circle be x2 + y2 = a2. Let the given external point be P (x1, y1).
From point P (x1, y1) two tangents PT and PR can be drawn to the circle, touching the circle at T and R respectively.
Then equation of pair of tangents is (x2 + y2a2) (x12 + y12a2) = (xx1 + yy1a2)2 or SS1 = T2, where S = x2 + y2a2, S1 = x12 + y12a2, and T = xx1 + yy1a2.

Director circle

The locus of the point of intersection of two perpendicular tangents to a given circle is a circle known as its director circle and is given by x2 + y2 = 2a2.
Equation of director circle for circle (xp)2 + (yq)2 = a2 is given by (xp)2 + (yq)2 = 2a2.

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