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Geometry - Circles and Tangents

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Question
SSCCGLENQA12043212
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A and B are centres of two circles of radii 11 cm and 6 cm, respectively. PQ is a direct common tangent to the circles. If AB with bar on top space equals space 13 space cm, then length of PQ with bar on top will be

  • 8.5 cm

  • 13 cm

  • 12 cm

  • 17 cm

Solution

C.

12 cm


Length of direct common tangent PQ:
PQ space equals space square root of AB squared minus left parenthesis straight r subscript 1 minus straight r subscript 2 right parenthesis squared end root space space space space space space equals space square root of 13 squared space minus space left parenthesis 11 minus 6 right parenthesis squared end root space space space space space space equals space square root of 13 squared space minus space 5 squared space end root space equals space square root of 169 space minus space 25 end root space equals space square root of 144 space equals space 12 space cm.

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Question
SSCCGLENQA12043303
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A and B are the centres of two circles with radii 11 cm and 6 cm respectively. A common tangent touches these circles at P and Q respectively. If AB  = 13 cm, then the length of PQ is

  • 13 cm

  • 17 cm

  • 8.5 cm

  • 12 cm

Solution

D.

12 cm


PQ space equals space square root of left parenthesis AB right parenthesis squared space minus space left parenthesis straight r subscript 1 minus straight r subscript 2 right parenthesis squared end root space space space space space space equals space square root of left parenthesis 13 right parenthesis squared minus left parenthesis 11 minus 6 right parenthesis squared end root space space space space space space equals space square root of 169 minus 25 end root space equals space square root of 144 space equals space 12 space cm.

Question
SSCCGLENQA12042336
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A chord of a circle is equal to its radius. A tangent is drawn to the circle at an extremity of the chord. The angle between the tangent and the chord is

  • 30°

  • 45°

  • 60°

  • 75°

Solution

A.

30°


OA = OB (radii of circle)
OA  = OB = AB (radii and chord are equal)
therefore increment AOB is a an equilateral triangle.
angle AOB space equals space angle OAB space equals space angle OBA space equals space 60 degree
angle OBC space equals space 90 degree
therefore space space angle ABD space equals space 180 degree minus angle OBC space minus angle OBA
rightwards double arrow space space angle ABD space equals space 180 degree minus 90 degree minus 60 degree rightwards double arrow space space angle ABD space equals space 180 degree minus 150 degree rightwards double arrow space space angle ABD space equals space 30 degree
 

Question
SSCCGLENQA12043428
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A circle has its centre at O. A tangent drawn from a point P, which is situated outside the circle, touches the circle at A. If PA= 4 cm and PO = 5 cm, then the length of the radius of the circle is

  • 1 cm

  • 2 cm

  • 3 cm

  • 4 cm

Solution

C.

3 cm

A circle has its centre at O. A tangent drawn from a point P, which is
In  Δ AOP,
     (OP)2 = (AP)2 + (AO)2
⟹   (5)2 = (4)2 + (AO)2
⟹    25 = 16  + (AO)2
⟹   (AO)2 = 9
∴     AO = 3 cm

Question
SSCCGLENQA12042929
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A, B, C are three points on a circle. The tangent at A meets BC produced at T, angle BTA space equals space 40 degree comma space space angle CAT space equals 44 degree. The angle subtended by BC at the centre of the circle is

  • 84°

  • 92°

  • 96°

  • 104°

Solution

D.

104°


angle CAT space equals space 44 degree space left parenthesis given right parenthesis angle BTA space equals space 40 degree space space left parenthesis given right parenthesis
angle ACT space equals space 180 degree space minus 44 degree space minus space 40 degree space equals space 96 degree angle CAT space equals space angle CBA space equals space 44 degree space space left parenthesis Alternate space segment space theorem right parenthesis
rightwards double arrow space space angle BCA space equals space 180 degree space minus space 96 degree space equals space 84 degree therefore space space angle BAC space equals space 180 degree space minus space 84 degree minus 44 degree space equals space 52 degree
∴  Angle subtended by BC at the centre = 2 x 52° = 104°

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