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The general equation of a conic has the form ๐ด๐ฅ squared plus ๐ต๐ฅ๐ฆ plus ๐ถ๐ฆ squared plus ๐ท๐ฅ plus ๐ธ๐ฆ plus ๐น equals zero.
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Consider the equation two ๐ฅ squared minus three ๐ฆ squared minus 16๐ฅ minus 30๐ฆ minus 49 equals zero.
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Calculate the value of the discriminant ๐ต squared minus four ๐ด๐ถ.
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Hence, identify the conic described by the equation.
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So the first thing we need to do in this question is identify our values.
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So first of all, we have ๐ด is equal to two.
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And thatโs because we have two ๐ฅ squared in our equation.
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And then we have ๐ต is equal to zero.
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And thatโs because we have no ๐ฅ๐ฆ term in our equation.
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And then we have ๐ถ is equal to negative three.
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And thatโs because we have negative three as the coefficient of ๐ฆ squared.
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Then ๐ท is equal to negative 16.
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And again, this is because the coefficient of ๐ฅ is negative 16.
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Then we have ๐ธ is equal to negative 30.
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And then, finally, we have ๐น is equal to negative 49.
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So weโve now found the values from ๐ด to ๐น.
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And one value that isnโt a surprise is ๐ต because ๐ต is equal to zero.
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And ๐ต is equal to zero is understandable because weโre told to calculate the value of the discriminant ๐ต squared minus four ๐ด๐ถ.
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And the discriminant is defined as ๐ต squared minus four ๐ด๐ถ when ๐ต is equal to zero.
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Okay, so now we have the values.
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Letโs use them to find the value of our discriminant.
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So to find the value of our discriminant, what weโre gonna do is utilize the values of ๐ด, ๐ต, and ๐ถ.
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So the way weโre gonna do that is by substituting in our values for ๐ด, ๐ต, and ๐ถ.
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And when we do that, what we get is zero squared โ thatโs cause ๐ต is equal to zero โ minus four multiplied by two multiplied by negative three.
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So thatโs gonna give us zero minus negative 24.
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We get that because zero squared is zero.
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And then weโve got four multiplied by two, which is eight, multiplied by negative three.
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Well, eight multiplied by three is 24.
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And a positive multiplied by a negative is a negative.
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So thatโs how we get our negative 24.
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So now weโve got zero minus negative 24, like I said.
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But if you subtract a negative, itโs the same as adding.
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So itโs the same as zero add 24.
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So therefore, weโve solved the problem because weโve got the value of the discriminant.
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And that value is 24.
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And we got that using the discriminant, which was ๐ต squared minus four ๐ด๐ถ.
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Okay, great, now letโs move on to the second part.
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And in the second part, what we need to do is identify the conic described by the equation.
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So to help us decide which conic is described, we have a set of conditions.
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The first of these is that if we have ๐ต squared minus four ๐ด๐ถ equal to zero and ๐ด or ๐ถ is equal to zero, then our conic is going to be a parabola.
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But if ๐ต squared minus four ๐ด๐ถ is less than zero and ๐ด is equal to ๐ถ, then the conic is going to be a circle.
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Then next we have if ๐ต squared minus four ๐ด๐ถ is less than zero again, but this time ๐ด does not equal ๐ถ, then this is going to be an ellipse.
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And then, finally, if weโve got ๐ต squared minus four ๐ด๐ถ is greater than zero, then weโre gonna have a conic which is a hyperbola.
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Okay, great, so we have all our conditions.
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Letโs use them to work out what our conic is.
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So therefore, using these conditions, we can say that our conic is going to be a hyperbola.
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And thatโs because ๐ต squared minus four ๐ด๐ถ is equal to 24.
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24 is greater than zero.
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And if we look at our conditions, itโs the final condition which Iโve met because it says ๐ต squared minus four ๐ด๐ถ is greater than zero.
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So therefore, we can confirm that the conic described by the equation two ๐ฅ squared minus three ๐ฆ squared minus 16๐ฅ minus 30๐ฆ minus 49 equals zero is a hyperbola.