Find the equations to the straight lines which are conjugate to the coordinate axes with respect to the conic A x^{2} + 2 H x y + B y^{2} = 1.Find the condition that they may coincide, and interpret the result.

Engineering

Mathematics

Line of Greatest Slope

Question

Find the equations to the straight lines which are conjugate to the coordinate axes with respect to the conic

A x^{2} + 2 H x y + B y^{2} = 1

.
Find the condition that they may coincide, and interpret the result.

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Given conic is

S = A x^{2} + 2 H x y + B y^{2} - 1 = 0

Partial differentiation of S w.r.t x is

2 A x + 2 H y = 0

---- equation 1

Partial Differentiation of S w.r.t y is

2 H x + 2 B y = 0

------equation 2

Both the equation 1 and 2 are conjugate to the coordinate axes

If the both equation of straight line coincide then there slope will be same .

Slope of equation 1 is

y = - (\frac{A}{H}) x

slope of equation 2 is

y = - (\frac{H}{B}) x

(\frac{A}{H}) = (\frac{H}{B})

Therefore

A B = H^{2}

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