Question

Determine the intersection of the hyperbolic paraboloid \(z=y^{2} / b^{2}-x^{2} / a^{2}\) with the plane \(b x+a y-z=0\). (Assume \(a, b>0 .)\)

Short answer

The intersection lines can be found by setting the hyperbolic paraboloid equation equal to the plane equation and solve for \(y\) and \(x\). After substituting and simplifying the equations, final values for \(x\) and \(y\) could be obtained.

Step-by-step solution

Express the hyperbolic paraboloid equation and the plane equation

The hyperbolic paraboloid is given by \(z=y^{2}/b^{2}-x^{2}/a^{2}\) and plane by \(bx+ay-z=0\). Our task is to find the intersection by setting both equations equal to each other and solving for \(x\) and \(y\).

Equating the equations

Setting the values of \(z\) from both equations to be equal, we obtain: \(y^{2}/b^{2}-x^{2}/a^{2} = bx+ay\). It is now in a form to solve for \(x\) and \(y\).

Isolate one variable and solve

Perform a manipulation to isolate any one variable, let's choose \(y\). So, \(y^{2}/b^{2} = x^{2}/a^{2} + bx+ay\). From here, solve for \(y\).

Find the value of x

Substitute the value of \(y\) from step 3 into the plane equation \(-bx+ay-z=0\) and solve for \(x\).

Final simplification

Use the results that have been obtained to simplify the equations and find the intersection.