Question

Find the coordinates of the ends of each latus rectum and equations of asymptotes.

Answer 1

For conic section of the form:

`((x^2)/(a^2))-((y^2)/(b^2))=1`

The Ends of the Lactus Rectum is given as:

`L=(ae,(b^2)/(a)),L=(ae,(-b^2)/(a))`

The e in the equation above is the Eccentricity of the Hyperbola.

This can be obtained by the formula:

`e=\frac{\sqrt[]{a^2+b^2}}{a}`

Thus, comparing the standard form of the conic with the given equation, we have:

`\begin{gathered} ((y+8)^2)/(16)-((x-3)^2)/(9)=1 \\ \text{This can be further expressed in the form:} \\ ((y+8)^2)/(4^2)-((x-3)^2)/(3^2)=1 \\ By\text{ comparing this with:} \\ (x^2)/(a^2)-(y^2)/(b^2)=1 \\ We\text{ can deduce that:} \\ a=4;b=3 \end{gathered}`

Then, we need to obtain the value of the Eccentiricity, e.

`\begin{gathered} e=\frac{\sqrt[]{a^2+b^2}}{a} \\ e=\frac{\sqrt[]{4^2+3^2}}{4} \\ e=\frac{\sqrt[]{16+9}}{4} \\ e=\frac{\sqrt[]{25}}{4}=(5)/(4) \end{gathered}`

Hence, the coordinate of the ends of the each lactus rectum is:

`\begin{gathered} L=(ae,(b^2)/(a)),L=(ae,(-b^2)/(a)_{}) \\ L=(4*(5)/(4),(3^2)/(4)),L=(4*(5)/(4),(-3^2)/(4)) \\ L=(5,(9)/(4)),L=(5,(9)/(4)) \end{gathered}`