Find domain and range of the slanted hyperbolaGiven the conic section Ax2+Bxy+Cy2+Dx+Ey+F=0 and I know that it is a hyperbola and B≠0.How to find its domain and range? I guess the method of Lagrange multipliers will fail here.

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anita1415snck · Accepted Answer

Explanation:To find a range, fix y and consider a quadratic equation with respect to x:Ax2+(By+D)x+C2y+Ey+F=0If&amp;nbsp;A=0&amp;nbsp;and&amp;nbsp;B=0, there is&amp;nbsp;x=−C2+Ey+FD&amp;nbsp;for each y and the domain is&amp;nbsp;(−∞,∞)(D≠0&amp;nbsp;, because, otherwise, the conic is degenerated).If&amp;nbsp;A=0&amp;nbsp;and&amp;nbsp;B≠0, there is no x for&amp;nbsp;y=−DB, for other values of y it is&amp;nbsp;x=−C2+Ey+FBy+D&amp;nbsp;and the domain is&amp;nbsp;(−∞,∞){−DB}.If&amp;nbsp;A≠0, qudratic equation has at least one solution, when its discriminant&amp;nbsp;d=(By+D)2−4A(C2+Ey+F)≥0.So, we need to find y, for which&amp;nbsp;(By+D)2−4A(C2+Ey+F)≥0.Rewrite&amp;nbsp;(B2−4AC)y2+(2BD−4AE)y+D2−4AF≥0If&amp;nbsp;B2−4AC=0, then1. If&amp;nbsp;2BD−4AE=0, the range is&amp;nbsp;(−∞,∞)&amp;nbsp;if&amp;nbsp;D2−4AF≥0&amp;nbsp;and&amp;nbsp;∅&amp;nbsp;otherwise.2. The range is&amp;nbsp;[−D2−4AF2BD−4AE,∞)&amp;nbsp;if&amp;nbsp;2BD−4AE&amp;gt;0.3. The range is&amp;nbsp;(−∞,−D2−4AF2BD−4AE]&amp;nbsp;if&amp;nbsp;2BD−4AE&amp;lt;0.The discriminant is&amp;nbsp;d1=(2BD−4AE)2−4(B2−4AC)(D2−4AF)1. If&amp;nbsp;d1&amp;lt;0, the range is&amp;nbsp;(−∞,∞)&amp;nbsp;if&amp;nbsp;B2−4AC&amp;gt;0&amp;nbsp;and&amp;nbsp;∅&amp;nbsp;otherwise.2. If&amp;nbsp;d1=0, the range is&amp;nbsp;(−∞,∞)&amp;nbsp;if&amp;nbsp;B2−4AC&amp;gt;0&amp;nbsp;and&amp;nbsp;{−2BD−4AE2(B2−4AC)}&amp;nbsp;otherwise.3. If&amp;nbsp;d1&amp;gt;0, there are two roots&amp;nbsp;y1=4AE−2BD−d12(B2−4AC)&amp;nbsp;and&amp;nbsp;y2=4AE−2BD+d12(B2−4AC). The range is&amp;nbsp;(−∞,y1]∪[y2,∞)&amp;nbsp;if&amp;nbsp;B2−4AC&amp;gt;0&amp;nbsp;and&amp;nbsp;[y1,y2]&amp;nbsp;otherwise.

Find domain and range of the slanted hyperbola Given

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