maths problem

1.
AC : CB = 1 : k
Then, BC : CA = k : 1

[4 + k(k - 1)]/(1 + k) = 2
4 + k2 - k = 2 + 2k
k2 - 3k + 2 = 0
(k - 1)(k - 2) = 0
k = 1 or k = 2

[k + k(-1)]/(1+k) = 0
This is an idenitity because L.S. = 0 and R.S. = 0

Hence, k = 1 or k = 2

2.
Let the y-intercept by c.
Then, the x-intercept = 2c

The lines passes through (0, c), (2c, 0) and (6, -1).

Equation of the line (point-slope form):
(y + 1)/(x - 6) = (c - 0)/(0 - 2c)
(y + 1)/(x - 6) = (1)/(-2)
x - 6 = -2y - 2
x + 2y - 4 = 0

3.
Radius
= (1/2)√[(a - 0)2 + (0 - b)2]
= (1/2)√(a2 + b2)

Centre
= ((a + 0)/2, (0 + (-b))/2)
= (a/2, -b/2)

Equation of the circle:
(x - a/2)2 + (y + b/2)2 = [(1/2)√(a2 + b2)]2
(4x - a)2 + (4y + b)2 = a2 + b2

When x = 0 and y = 0:
L.S. = a2 + b2 = L.S.
Hence, (0, 0) lies on the circle.

When x = a and y = b
L.S. = 9a2 + 25b2 ≠ L.S.
Hence, (a, b) does not lie on the circle.

When x = a and y = -b
L.S. = 9a2 + 9b2 ≠ L.S.
Hence, (a, -b) does not lie on the circle.

4.
Centre of the circle = (0, 0)

Slope of the line joining the centre and (1, -2)
= (0 + 2)/(0 - 1)
= -2

The chord is perpendicular to the above line.
Slope of the chord
= -1/(-2)
= 1/2

Equation of the chord:
(y + 2)/(x - 1) = 1/2
x - 1 = 2y + 4
x - 2y - 5 = 0

5.
The answer is C.

Centre of the circle, O = (0, 0)
Radius of the circle, r = √10

The distance between a tangent and the centre of a circle must be equal to the radius of the circle.

A is incorrect.
Distance between the line and the centre (0, 0)
= |(0) + 5(0) + 8|/√[12 + 52]
≠ radius

B is incorrect.
Distance between the line and the centre
= |2(0) - 3(0) - 7|/√[22 + (-3)2]
≠ radius

C is correct.
Distance between the line and the centre
= |3(0) - (0) + 10|/√[32 + (-1)2]
= 10/√10
= √10
= radius

D is incorrect.
Distance between the line and the centre
= 10 - 0
= 10
≠ radius

E is incorrect.
Distance between the line and (0, 0)
= |(0) - (0) - 2|/√[12 + 12]
≠ radius
=

too late

2.Find the equation of the line passing through (6,-1) and with x-intercept twice that of the y-intercept.
sol
a:x-intercept
b:y-intercept
a=2b
x/a+y/b=1
x/(2b)+y/b=1
x+2y=2b
6-2=2b
b=2
x+2y=4