Question

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−1)2−(y−2)2=3

Accepted Answer

Identify the center of the hyperbola by rewriting the equation in the form $$\frac{(x-h)^2$}{a^2$} - \frac{(y-k)^2$}{b^2$} = 1. $$Here, the center is at $$(h, k). $$Start by dividing both sides of the equation $$ (x-1)^2$ - (y-2)^2$ = 3  by 3 to $$get it into standard form. From the standard form, determine $$a^2$$ and $$b^2. $$The term with the positive sign corresponds to $$\frac{(x-h)^2$}{a^2$}$$, and the term with the negative sign corresponds to $$\frac{(y-k)^2$}{b^2$}. $$Identify $$a$$ and $$b by $$taking square roots of $$a^2$$ and $$b^2$$ respectively. Locate the vertices of the hyperbola. Since the $$x-$$term is positive, the transverse axis is horizontal. The vertices are located $$a$$ units left and right from the center along the $$x-$$axis, at points $$(h \pm a, k). $$Find the foci of the hyperbola. Use the relationship $$c^2$$ = $$a^2$$ + $$b^2 to $$find $$c$$, where $$c is $$the distance from the center to each focus along the transverse axis. The foci are at $$(h \pm c, k). $$Write the equations of the asymptotes. For a hyperbola centered at $$(h, k)$$ with a horizontal transverse axis, the asymptotes are given by $$y - k = \pm \frac{b}{a} (x - h). $$Substitute the values of $$a$$, $$b$$, $$h$$, and $$k to $$get the asymptote equations.

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. (x−1)²−(y−2)²=3

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Concetti chiave

Standard Form of a Hyperbola

Foci and Vertices of a Hyperbola

Equations of the Asymptotes