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Intersection between Tangent and Secant

Suppose two c...

Question

Suppose two circles touching at A. Through A two lines are drawn intersecting one circle in P, Q and other circle in X,Y.
Prove that $PQ ∥ XY$

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Solution

$Construction : Join O' Q, O' A, OX, OA, PQ and XY . Here, ∠ O' AW = 90 ° and ∠ OAW = 90 ° (Radius is perpendicular to tangent) So, ∠ O' AW + ∠ OAW = 180 ° Hence, OO' is a straight line . O' A = O' Q (Radii of the circle) i . e ., ∠ 1 = ∠ 2 . . . (1) (Angle opposite to equal sides) OA = OX (Radii of the circle) i . e ., ∠ 4 = ∠ 5 . . . (2) (Angle opposite to equal sides) In ∆ O' QA, we have : ∠ 1 + ∠ 2 + ∠ 3 = 180 ° \Rightarrow ∠ 1 + ∠ 1 + ∠ 3 = 180 ° [Using equation (1)] \Rightarrow ∠ 3 = 180 ° - 2 ∠ 1 . . . (3) In ∆ OXA, we have : ∠ 4 + ∠ 5 + ∠ 6 = 180 ° \Rightarrow ∠ 4 + ∠ 4 + ∠ 6 = 180 ° [Using equation (2)] \Rightarrow ∠ 6 = 180 ° - 2 ∠ 4 . . . (4) But ∠ 1 = ∠ 4 (Vertically opposite angle) Hence, ∠ 3 = ∠ 6 Dividing both sides by 2, we get : \frac{1}{2} ∠ 3 = \frac{1}{2} ∠ 6 The angle subtended by the arc at the centre is twice the angle subtended by the arc on any part of the circle . ∴ ∠ QPA = ∠ XYA$

Since alternate angles are equal, $PQ ∥ XY$ .

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Suppose two circles touching at A. Through A two lines are drawn intersecting one circle in P, Q and other circle in X,Y. Prove that PQ∥XY

Suppose two circles touching at A. Through A two lines are drawn intersecting one circle in P, Q and other circle in X,Y.
Prove that $PQ ∥ XY$