If the pair of lines represented by a x^2+2 h x y+b y^2+2 g...

If the pair of lines represented by $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ intersect on the y axis, then
(a) $a b c-f g h=b g^{2}+c h^{2}$
(b) $a b c+2 f g h=b g^{2}+c h^{2}$
(c) $a b c+f g h=a f^{2}+c h^{2}$
(d) $a b c=b g^{2}+c h^{2}$

Key Concepts

General Second Degree Equation Representing Two Lines

A second degree equation in x and y, such as Ax² + 2Hxy + By² + 2Gx + 2Fy + C = 0, can represent a pair of straight lines when the quadratic factors into two linear factors. This form is fundamental in coordinate geometry since it unifies the description of conic sections and degenerate cases (like intersecting lines) under one umbrella.

Point of Intersection of Lines from a Conic Equation

For a quadratic equation representing a pair of lines, the point where the lines intersect—their point of concurrence—can be found by solving the linear system obtained from the partial derivatives of the equation with respect to x and y. This method yields the coordinates of the intersection point in terms of the equation's coefficients.

Intersection of Lines on a Specific Axis

When the intersection point of the two lines lies on a particular axis, such as the y-axis, the x-coordinate of the point must be zero. Imposing this condition on the intersection point provides a relationship among the coefficients of the given quadratic equation. This concept is widely used to deduce specific coefficient constraints for geometric alignment of the lines.

Coefficient Relationships and Geometric Conditions

The coefficients in the general quadratic equation are interrelated such that certain geometric conditions—in this case, the intersection point lying on the y-axis—impose additional constraints. Deriving these relationships often involves setting specific coordinate conditions and equating expressions obtained from the general formulas for the point of intersection.

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