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5. Find the length of the chord cut off from the line 2x + 3y = 13 by the circle
x2 + y2 = 26
6. Find the coordinates of the points of intersection of the line x + 2y = 6 with the circle:
x2 + y2 - 2x - 2y - 39 = 0
7. Find equations of the tangents to the circle x2 + y2 = 2
(i) parallel to the line x - 2y + 1 = 0
(ii) perpendicular to the line 3x + 2y = 6
8. Find equations of the tangents drawn from
(i) (0 , 5) to x2 + y2 = 16
(ii) (-1 ,2 ) to x2 + y2 + 4x + 2y = 0
(iii) (-7, -2 ) to (x + 1)2 + (y - 2)2 = 26
Also find the points of contact
9. Find an equation of the chord of contact of the tangents drawn from (4 , 5) to the circle
2x2 + 2y2 - 8x + 12y + 21 = 0

6.3 ANALYTIC PROOFS OF IMPORTANT
PROPERTIES OF A CIRCLE

A line segment whose end points lie on a circle is called a chord of the circle. A diameter
of a circle is a chord containing the centre of the circle.

Theorem: Length of a diameter of the circle x2 + y2 = a2 is 2a.

Proof: Let AOB be a diameter of the circle
x2 + y2 = a2 (1)
O(0,0) is center of (1).
Let the coordinates of A be (x1, y1).
Equation of AOB is

y = y1 x (2)
x1

Substituting the value of y from (2) into (1), we have

version: 1.1

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