Homogeneous Equations and Quadratics

The video tutorial explains how to solve a system of equations involving X and Y. It begins by introducing the problem and the given equations. The tutorial then demonstrates how to transform these equations into a homogeneous form. By solving a quadratic equation for M, the video derives possible values for X and Y. Finally, it concludes with the solutions, providing four possible pairs of values for X and Y.

What are the given equations in the problem?

• X^2 - 2Y^2 + 5XY = 26 and X^2 - Y^2 - XY = 37

• X^2 + 2Y^2 - 5XY = -26 and X^2 + Y^2 + XY = 37

• X^2 + Y^2 = 26 and X^2 + 2Y^2 = 37

• X^2 + 2Y^2 + 5XY = 26 and X^2 + Y^2 - XY = 37

Why is the equation X^2 + Y^2 + XY = 37 not homogeneous?

• The degrees of X and Y terms are different.

• The degree of the constant term is not zero.

• The equation has a constant term.

• The equation is already homogeneous.

How can the equation X^2 + 2Y^2 - 5XY = -26 be made homogeneous?

• By dividing by -26

• By adding 26 to both sides

• By subtracting 26 from both sides

• By multiplying by 26

What is the result of cross-multiplying the homogeneous equations?

• 37X^2 - 74Y^2 + 185XY = 26X^2 - 26Y^2 + 26XY

• 37X^2 + 74Y^2 + 185XY = 26X^2 + 26Y^2 + 26XY

• 37X^2 + 74Y^2 - 185XY = 26X^2 + 26Y^2 + 26XY

• 37X^2 + 74Y^2 - 185XY = -26X^2 - 26Y^2 - 26XY

What substitution is made to simplify the homogeneous equation?

• Y = MX

• X = Y/M

• X = MY

• Y = X/M

What is the quadratic equation derived in terms of M?

• 100M^2 + 63M - 159 = 0

• 100M^2 + 159M - 63 = 0

• 100M^2 - 159M + 63 = 0

• 100M^2 - 63M + 159 = 0

What is the discriminant (D) of the quadratic equation?

• 159^2

• 9^2

• 100^2

• 63^2

What are the possible values of M?

• 3/4 and 21/25

• 3/5 and 21/20

• 5/3 and 20/21

• 4/3 and 25/21

What are the final solutions for X and Y?

• (3, 4), (-3, -4), (25/√43, 21/√43), (-25/√43, -21/√43)

• (4, 3), (-4, -3), (25/√43, 21/√43), (-25/√43, -21/√43)

• (3, 4), (-3, -4), (21/√43, 25/√43), (-21/√43, -25/√43)

• (4, 3), (-4, -3), (21/√43, 25/√43), (-21/√43, -25/√43)