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Find the equations of the straight lines passing through the intersection point of the lines x - 2y - 5 = 0 and 7x + y = 50, which cut the circle x² + y² = 100 into two arcs whose lengths are in the ratio 2: 1.

1. 3x + 4y = 25 and 4x - 3y = 25
2. 3x - 4y = 25 and 4x + 3y = 25
3. x + y = 8 and x - y = 6
4. 3x + 4y = 50 and 4x - 3y = 50

Correct answer: 3x + 4y = 25 and 4x - 3y = 25

Solution

The two lines meet at (7, 1). A 2:1 arc split needs a 120 deg central angle, giving a chord at distance 10*cos(60 deg) = 5 from the centre. The two required lines through (7,1) at distance 5 from the origin are 3x + 4y = 25 and 4x - 3y = 25.

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