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Length of Chord of Contact

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Qonsider a family of circles passing through the point { of intersection of the lines \sqrt3(y-1)=x-1 and y-1{=\sqrt3(x-1) and having its centre on the acute angle bisector of the given lines. Then the common chords of { each member of the family and the circle x^2+y^2+4x-{6y+5=0 are concurrent at

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Q. Consider a family of circles, each passing through (1, 1) and having its center on the bisector of the acute angle between the lines

\sqrt{3} (y - 1) = x - 1

y - 1 = \sqrt{3} (x - 1)

. Prove that the common chords of each member of the family and circle

x_{2} + y_{2} + 4 x - 4 y - 1 = 0

are concurrent. Find the point of concurrency.

Q. Let any circle

S

passes through the point of intersection of lines

\sqrt{3} (y - 1) = x - 1

y - 1 = \sqrt{3} (x - 1)

and having its centre on the acute angle bisector of the given lines. If the common chord of

S

x^{2} + y^{2} + 4 x - 6 y + 5 = 0

passes through a fixed point, then the fixed point is

Q. Let any circle

S

passes through the point of intersection of lines

\sqrt{3} (y - 1) = x - 1

y - 1 = \sqrt{3} (x - 1)

and having its centre on the acute angle bisector of the given lines. If the common chord of

S

x^{2} + y^{2} + 4 x - 6 y + 5 = 0

passes through a fixed point, then the fixed point is

Q. Consider a family of circles passing through the points

(3, 7)

(6, 5)

. The chords in which the circle

x^{2} + y^{2} - 4 x - 6 y - 3 = 0

cuts the family of circles are concurrent at the point

Q. Consider a family of circles passing through two fixed points

A (3, 7)

B (6, 5)

. The chord, in which the circle

x^{2} + y^{2} - 4 x - 6 y - 3 = 0

cuts each member of the family of circles, passes through a fixed point

(a, b)

. Find the numerical value of

a + 3 b

.

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