Video: Identifying Conic Sections with the Discriminant

The general equation of a conic has the form ๐ด๐‘ฅยฒ + B๐‘ฅ๐‘ฆ + C๐‘ฆยฒ + ๐ท๐‘ฅ + ๐ธ๐‘ฆ + ๐น = 0. Consider the equation 2๐‘ฅยฒ โˆ’ 3๐‘ฆยฒ โˆ’16๐‘ฅ โˆ’ 30๐‘ฆ โˆ’ 49 = 0. Calculate the value of the discriminant ๐ตยฒ โˆ’ 4๐ด๐ถ. Hence, identify the conic described by the equation.

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Video Transcript

The general equation of a conic has the form ๐ด๐‘ฅ squared plus ๐ต๐‘ฅ๐‘ฆ plus ๐ถ๐‘ฆ squared plus ๐ท๐‘ฅ plus ๐ธ๐‘ฆ plus ๐น equals zero. Consider the equation two ๐‘ฅ squared minus three ๐‘ฆ squared minus 16๐‘ฅ minus 30๐‘ฆ minus 49 equals zero. Calculate the value of the discriminant ๐ต squared minus four ๐ด๐ถ. Hence, identify the conic described by the equation.

So the first thing we need to do in this question is identify our values. So first of all, we have ๐ด is equal to two. And thatโ€™s because we have two ๐‘ฅ squared in our equation. And then we have ๐ต is equal to zero. And thatโ€™s because we have no ๐‘ฅ๐‘ฆ term in our equation. And then we have ๐ถ is equal to negative three. And thatโ€™s because we have negative three as the coefficient of ๐‘ฆ squared. Then ๐ท is equal to negative 16. And again, this is because the coefficient of ๐‘ฅ is negative 16. Then we have ๐ธ is equal to negative 30. And then, finally, we have ๐น is equal to negative 49.

So weโ€™ve now found the values from ๐ด to ๐น. And one value that isnโ€™t a surprise is ๐ต because ๐ต is equal to zero. And ๐ต is equal to zero is understandable because weโ€™re told to calculate the value of the discriminant ๐ต squared minus four ๐ด๐ถ. And the discriminant is defined as ๐ต squared minus four ๐ด๐ถ when ๐ต is equal to zero.

Okay, so now we have the values. Letโ€™s use them to find the value of our discriminant. So to find the value of our discriminant, what weโ€™re gonna do is utilize the values of ๐ด, ๐ต, and ๐ถ. So the way weโ€™re gonna do that is by substituting in our values for ๐ด, ๐ต, and ๐ถ. And when we do that, what we get is zero squared โ€” thatโ€™s cause ๐ต is equal to zero โ€” minus four multiplied by two multiplied by negative three. So thatโ€™s gonna give us zero minus negative 24. We get that because zero squared is zero. And then weโ€™ve got four multiplied by two, which is eight, multiplied by negative three. Well, eight multiplied by three is 24. And a positive multiplied by a negative is a negative. So thatโ€™s how we get our negative 24.

So now weโ€™ve got zero minus negative 24, like I said. But if you subtract a negative, itโ€™s the same as adding. So itโ€™s the same as zero add 24. So therefore, weโ€™ve solved the problem because weโ€™ve got the value of the discriminant. And that value is 24. And we got that using the discriminant, which was ๐ต squared minus four ๐ด๐ถ.

Okay, great, now letโ€™s move on to the second part. And in the second part, what we need to do is identify the conic described by the equation. So to help us decide which conic is described, we have a set of conditions. The first of these is that if we have ๐ต squared minus four ๐ด๐ถ equal to zero and ๐ด or ๐ถ is equal to zero, then our conic is going to be a parabola. But if ๐ต squared minus four ๐ด๐ถ is less than zero and ๐ด is equal to ๐ถ, then the conic is going to be a circle.

Then next we have if ๐ต squared minus four ๐ด๐ถ is less than zero again, but this time ๐ด does not equal ๐ถ, then this is going to be an ellipse. And then, finally, if weโ€™ve got ๐ต squared minus four ๐ด๐ถ is greater than zero, then weโ€™re gonna have a conic which is a hyperbola.

Okay, great, so we have all our conditions. Letโ€™s use them to work out what our conic is. So therefore, using these conditions, we can say that our conic is going to be a hyperbola. And thatโ€™s because ๐ต squared minus four ๐ด๐ถ is equal to 24. 24 is greater than zero. And if we look at our conditions, itโ€™s the final condition which Iโ€™ve met because it says ๐ต squared minus four ๐ด๐ถ is greater than zero.

So therefore, we can confirm that the conic described by the equation two ๐‘ฅ squared minus three ๐‘ฆ squared minus 16๐‘ฅ minus 30๐‘ฆ minus 49 equals zero is a hyperbola.

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