Question

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins (in centimeters) are respectively 'a' and 'b'. Find the numerical value of a²+ b².

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{ellipse}\mathrm{be}\frac{{\mathrm{x}}^2}{{\mathrm{A}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{B}}^2}=1\left(\mathrm{A}>\mathrm{B}\right). \\ \mathrm{Let}{\mathrm{F}}_1\mathrm{and}{\mathrm{F}}_2\mathrm{be}\mathrm{the}\mathrm{foci}\mathrm{of}\mathrm{the}\mathrm{ellipse}.\mathrm{Suppose}\mathrm{P}\mathrm{is}\mathrm{any}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{ellipse}. \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{major}\mathrm{axis}=6 \\ \Rightarrow 2\mathrm{A}=6 \\ \Rightarrow \mathrm{A}=3 \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{minor}\mathrm{axis}=4 \\ \Rightarrow 2\mathrm{B}=4 \\ \Rightarrow \mathrm{B}=2 \\ \mathrm{Let}\mathrm{e}\mathrm{be}\mathrm{the}\mathrm{eccentricity}\mathrm{of}\mathrm{the}\mathrm{ellipse}. \\ {\mathrm{B}}^2={\mathrm{A}}^2\left(1-{\mathrm{e}}^2\right) \\ \Rightarrow \mathrm{e}=\sqrt{1-\frac{{\mathrm{B}}^2}{{\mathrm{A}}^2}}=\sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{3} \\ \mathrm{Distance}\mathrm{between}\mathrm{the}\mathrm{foci}=\mathrm{b}=2\mathrm{Ae}=2\times 3\times \frac{\sqrt{5}}{3}=2\sqrt{5} \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{string}\mathrm{required}\mathrm{to}\mathrm{describe}\mathrm{the}\mathrm{ellipse}=\mathrm{a} \\ \Rightarrow {\mathrm{PF}}_1+{\mathrm{PF}}_2=\mathrm{a} \\ \Rightarrow 2\mathrm{A}=\mathrm{a} \\ \Rightarrow \mathrm{a}=2\times 3=6 \\ \therefore {\mathrm{a}}^2+{\mathrm{b}}^2={\left(6\right)}^2+{\left(2\sqrt{5}\right)}^2=36+20=56 \end{gathered}$$