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Coordinate geometry mathematics class note circles 5 (2)

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NOTE M A T H E M A T I C S CIRCLES CHORD BISECTED AT A POINT AND COMMON TANGENTS What you already know • • • • Number of tangents on a circle from a point Pair of tangents on a circle from a point Length of a tangent Power of a point • • • • Angle between a pair of tangents Director circle Chord of contact Segments of secants • Common tangents to two given circles What you will learn • • Chord bisected at a given point Pole and Polar Chord Bisected at a Given Point S=0 The equation of the chord of the circle S = 0, whose midpoint is P( x1 , y1 ), is T = S1. T = S1 C B P (x1, y1) Proof Circle S: x2 + y2 + 2gx + 2fy + c = 0 Centre C ≡ (-g, -f) Now, let AB be the chord whose midpoint is P(x1, y1). ⇒ AB ⟂ CP ⇒ Slope of CP × Slope of AB = -1 y1 + f We know that, the slope of CP = x1 + g © 2020, BYJU'S. All rights reserved A 02 x + g ∴ Slope of AB = -  1   y1 + f  Now, the equation of AB is as follows: x +g ⇒(y - y1) = - 1 (x - x1) y1 + f After simplifying, we get, ⇒ xx1 + yy1 + gx + fy = x12 + y12 + gx1 + fy1 A Adding gx1 + fy1 + c on both sides, we get, ⇒ xx1 + yy1 + gx + gx1 + fy + fy1 + c = x12 + y12 + 2gx1 + 2fy1 + c Arranging the given terms on both sides, we get, 2g(x + x1) 2f(y + y1) ⇒ xx1 + yy1 + + + c = x12 + y12 + 2gx1 + 2fy1 + c 2 2 Finally, we get, C P (x1, y1) B ⇒T = S1 Note S=0 The length of the chord of the circle S = 0, whose midpoint is P( x1, y1), is 2 |S1|. B |S 1| 2√ P (x ...
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