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Direct common Tangents and Transverse common Tangents

  1. Transverse common tangent In the figure given below, PQ and RS are the transverse common tangents. Transverse common tangents intersect the line joining the centre of the two circles. They divide the line in the ratio r1 : r2.
    AB : BC = r1 : r2
    Assume AC = Distance between centres = d
    Description: P-341-1.tif
    PQ2 = RS2 = d2 – (r1 + r2)2
  2. Direct common tangent
    Description: P-341-2.tif
    In the figure given above, PQ and RS are direct common tangents.
Points A and C are the point of tangency for the first circle and similarly points B and D are the point of tangency for the second circle. AB and CD are known as lengths of the direct common tangents and they will be same.
CD2 = AB2 = d2 – (r1 – r2)2


Description: P-341-3.tif
In the figure given above, AB is a tangent and ACD is a secants
  1. AB2 = AC × AD
  2. AE × AF = AC × AD

Important theorems related to Circle

  1. If C is the mid-point of AB, then OC is perpen­dicular to AB. And vice versa is also true.
Description: P-341-4.tif
  1. Angles in the same segment will be equal.
Description: P-341-5.tif
In the figure given above, 
a = b.
  1. Angle subtended by a chord at the centre is two times the angle subtended on the circle on the same side. In the figure given below, 2a = 2b = c.
Description: P-341-6.tif
  1. Angle subtended by a diameter of the circle is a right angle.
  2. Alternate segment theorem
Description: P-342-1.tif
In the figure above, AT is the tangent. a = Alternate segment b c = Alternate segment d
  1. Converse of alternate segment theorem If a line is drawn through an end point of a chord of a circle so that the angle formed by it with the chord is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle.
Description: P-342-2.tif
AB is a chord of a circle and a line PAQ such that BAQ = ACB, where C is any point in the alter­nate segment ACB, then PAQ is a tangent to the circle.
  1. Tangent drawn to a circle from a point are same in length.
    In the figure below, tangents are drawn to the circle from point A and AT1 and AT2 are the tangents.
    1. AT1 = AT2 ii. 1 = 2
    2. AT1 2 + OT12 = AT2 2 + OT22 = AO2
Description: P-342-3.tif

Cyclic Quadrilateral

Consider the figure given below:
Description: P-342-4.tif
If we have a b = 180° and quadrilateral AXBP has all its vertices on a circle, then such a quadrilateral is called a Cyclic quadrilateral.
For a cyclic quadrilateral, the sum of the opposite angles of a quadrilateral in a circle is 180°.
It can also be seen that exterior CBE = internal ADC = 180° −ABC.
Description: P-342-5.tif

Using Brahmagupta’s formula to find out the area of a cyclic quadrilateral We know that A + B = p So, area of cyclic quadrilateral
Description: 5771.png 
Where terms used are having their meaning.
[Cos 90° = 0]

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