This problem were given six equations for surfaces and we want to identify what kind of quadratic surfaces they are. So here I've written the six different forums of quadratic surfaces and kind of with general coefficients like a BNC. And so we'll start with the first one. So we're kind of familiarizing ourselves with the different quadratic surfaces. So we have six X squared plus three y squared plus four z squared is equal to 12. So first, what we want to dio is when we have a second order terms in X, Y and Z, we want to normalize the right hand side. So we'll divide through by 12. We have one is equal to X squared over two plus y squared over four plus Z squared over three. And so this looks like an ellipse oId with a equal to the square to to be equal to two and C equal to square to three. Next up, we have y squared minus X squared minus Z is equal to zero. So noticed that Z is a first order term. So we're gonna reorganize this with Z is the first element, and we're also gonna make its coefficient one sold, but we'll multiply through by negative one. So we have Z plus X squared minus y squared is equal to zero. This looks like a hyperbolic crab Lloyd with a equal toe one and be equal to one. And next problem. We have nine x squared plus y squared minus nine. Z squared is equal to nine. So again we have a second order terms on the left hand side and a constant on the right hand side. So let's divide through by that constant to normalize, it will have one is equal. Thio X squared plus y squared minus. So sorry y squared over nine minus c squared. And since there's one negative term on the left hand side, this is gonna be a hyper high purple Lloyd of one sheet. So then we have a is equal toe. One b is equal to three and C is equal to one. Next, we have four x squared plus y squared minus four. Z squared is equal to negative four. So we're gonna normalize the left hand side started the right hand side, and so we have one is equal to so negative X squared minus weiss Word over four plus z squared. We have to negative terms. Therefore, this is a hyper Boyd of two sheets with a equal to one. Be equal to two and C equal to one. Next we have to Z minus X squared minus y squared. Sorry. So minus X squared minus four. Why squared is equal to zero. So first, what we want to do is we want to normalize the coefficient. Is he still have Z minus X squared over two minus two. Y squared is equal to zero. We have a linear term and to negative terms. So this is gonna be and elliptical Prabal Lloyd with a equal to square tissue and be equal to square of one half. Or we could write that as, um or to to over two, if you like. Next, we have 12 z squared minus three x squared is equal toe four y squared. So here we have all second order terms. However, we have no constant on the right hand side. So this is an elliptical cone. We'll move everything to the left hand side and then we want to normalize the coefficient, etc. So we have Z squared. So we're gonna be dividing through by 12. So is the squared minus X squared over four minus y squared over three is equal to zero. So then a is equal to two and B is equal to score three and that completes this problem.