By calculating the discriminant, identify the type of conic that is described by the equation 𝑥 squared plus 𝑦 squared plus 10𝑥 minus four 𝑦 plus 28 equals zero.
Well, in this question, what we’re are asked to do is use the discriminant to identify the type of conic. Well, to find the discriminant, if we have the equation in the form 𝐴𝑥 squared plus 𝐵𝑥𝑦 plus 𝐶𝑦 squared plus 𝐷𝑥 plus 𝐸𝑦 plus 𝐹 equals zero, where 𝐵 equals zero, the discriminant is equal to 𝐵 squared minus four 𝐴𝐶.
Well, to check that we can use that for our question, we’ve checked each of the conditions. So we’ve got an 𝑥 squared. We’ve got 𝑦 squared. We’ve got 𝑥. We’ve got 𝑦. And then, we’ve got a constant and also 𝐵, which is 𝑥𝑦. It’s gonna be equal to zero because 𝐵 is the coefficient of 𝑥𝑦. And we don’t have any 𝑥𝑦 term. So it would be zero.
So if we take a look at our equation, what we need to do is find out what 𝐴, 𝐵, and 𝐶 are, to be able to work out our discriminant. Well, we know that 𝐵 is equal to zero. And 𝐴 is gonna be equal to one because 𝐴 is the coefficient of 𝑥 squared. And then, we’ve got a single 𝑥 squared. So that have a coefficient of one. And 𝐶 is also gonna be equal to one because 𝐶 is a coefficient of 𝑦 squared. And we can see that we’ve got a signle 𝑦 squared. So again that have a coefficient of one.
So now, we’re able to work out the discriminant. But before we do that, why we’re going to do that? How is that gonna help us to identify the conic that’s described by our equation.
Well, we have a set of general rules for the determinant of an equation that’s conic. And that is that if we’ve 𝐵 squared minus four 𝐴𝐶 is equal to zero and 𝐴 or 𝐶 is equal to zero, it’s going to be a parabola. If it is less than zero and 𝐴 is equal to 𝐶, then it’s a circle. If it’s less than zero and 𝐴 is not equal to 𝐶, then it’s an ellipse. And if it’s greater than zero, then it’s gonna be a hyperbola.
So if we substitute our values for 𝐴, 𝐵, and 𝐶 into our discriminant, we’re going to get zero squared minus four multiplied by one multiplied by one, which is gonna give us zero minus four, which is equal to negative four. Well, negative four is less than zero. And 𝐴 is equal to 𝐶. So this fulfills the conditions for the second rule.
So therefore, we can say that, by using the discriminant, we’ve identified the type of conic that is described by the equation 𝑥 squared plus 𝑦 squared plus 10𝑥 minus four 𝑦 plus 28 equals zero. And we’ve identified that it is a circle. And that is because 𝐵 squared minus four 𝐴𝐶, the discriminant, is less than zero and 𝐴 is equal to 𝐶.