If k is positive in the hyperbola equation, where are the branches located?

In the hyperbola equation, when we say that ( k ) is positive, it generally indicates the standard form of the hyperbola centered at the origin, which can be expressed as ( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 ). In this format, the branches of the hyperbola open towards the left and right.

Given that ( k ) is positive, the branches will reside in the first and third quadrants. This is because in the first quadrant, both ( x ) and ( y ) are positive, allowing for valid solutions where the equation holds true. Similarly, in the third quadrant, both ( x ) and ( y ) are negative, which also satisfies the equation. The other quadrants (second and fourth) would not contain any parts of the hyperbola as they involve computations where either ( x ) or ( y ) would not yield valid (real number) solutions due to the structure of the equation.

Thus, the correct choice reflects that the branches of the hyperbola, when ( k ) is positive, are indeed located in the first and third quadrants.