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Equation of the bisector of the angle between two lines

This type of question, asked 3 times: 30 Jan 2024, 24 Jan 2026 S1, 4 Apr 2026 S1. All 3 questions below, oldest first.

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  1. Question 1

    30 Jan 2024

    157
    If x2−y2+2hxy+2gx+2fy+c=0x^{2}-y^{2}+ 2hxy + 2gx + 2fy + c = 0 is the locus of a point, which moves such that it is always equidistant from the lines x+2y+7=0x + 2y + 7 = 0 and 2x−y+8=02x - y + 8 = 0, then the value of g+c+h−fg + c + h - f equals

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  2. Question 2

    24 Jan 2026 S1

    63
    Let A(1,0),B(2,−1)A(1,0),B(2, - 1) and C(73,43)C\left( \frac{7}{3},\frac{4}{3} \right) be three points. If the equation of the bisector of the angle ABC is αx+βy=5\alpha x + \beta y = 5, then the value of α2+β2\alpha^{2}+\beta^{2} is

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  3. Question 3

    4 Apr 2026 S1

    61
    Let the line L1:x+3=0L_{1}:x + 3 = 0 intersect the lines L2:x−y=0L_{2}:x - y = 0 and L3:3x+y=0L_{3}:3x + y = 0 at the points A and B , respectively. Let the bisector of the obtuse angle between the lines L2L_{2} and L3L_{3} intersect the line L1L_{1} at the point C . Then BC2:AC2BC^{2}:AC^{2} is equal to :

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