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Standard X

Mathematics

Two Circles Touching Internally and Externally

A straight li...

Question

A straight line drawn through the point of contact of two circles with centres A and B intersect the circles at P and Q respectively. Show that AP and BQ are parallel.

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Solution

$M$ is the point of contact of two circles touching externally.
In $△ A P M, A M = A P$ (radii of $G$ )
$∠ A P M = ∠ A M P$ (angle opposite to equal side)
$∠ A M P = ∠ Q M B$ (opposite angle) $(1)$

In $△ M B Q, M B = B Q = r a d i i$
$∠ Q M B = ∠ B Q M$ (angle opposite to equal side) $(2)$

From $(1)$ and $(2)$ , we get $∠ A P M = ∠ B Q M$
Hence, $A P ∥ B Q$ as ( $∠ A P M, ∠ B Q M$ are alternate interior angles)

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Two circles with centres $O$ and $O^{'}$ intersect at two points $A$ and $B$ . $A$ line $P Q$ is drawn parallel to $O O^{'}$ through $A$ or $B$ , intersecting the circles at $P$ and $Q .$ Prove that $P Q = 2 O O^{'} .$

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Question 18
Two circles with centres O and O' intersect at two points A and B. A line PQ is drawn parallel to OO' through A (or B) intersecting the circles at P and Q. prove that PQ = 2 OO'.

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