Question

if the distance between the points P(a cos48⁰ ,0) and Q(0, a cos 12⁰) is d,then d² - a²=
1.a²/4 (root 5 -1)
2.a²/4 (root 5 +1)
3.a/8 (root 5 -1)
4.​a²/8 (root 5 +1)

Answer 1

$$\begin{gathered} \mathrm{P}\left(\mathrm{a}\cos 48°,0\right)\mathrm{and}\mathrm{Q}\left(0,\mathrm{a}\cos 12°\right) \\ {\mathrm{d}}^2={\mathrm{PQ}}^2={\left(\mathrm{a}\cos 12°-0\right)}^2+{\left(0-\mathrm{a}\cos 48°\right)}^2 \\ ={\mathrm{a}}^2{\cos }^{2}12°+{\mathrm{a}}^2{\cos }^{2}48° \\ ={\mathrm{a}}^2\left({\cos }^{2}12°+{\cos }^{2}48°\right) \\ ={\mathrm{a}}^2\left(\frac{1+\cos 24°}{2}+\frac{1+\cos 96°}{2}\right) \\ ={\mathrm{a}}^2\left(1+\cos 24°+\cos 96°\right) \\ {\mathrm{d}}^2-{\mathrm{a}}^2={\mathrm{a}}^2\left(1+0.913-0.104\right)-{\mathrm{a}}^2 \\ =0.8084{\mathrm{a}}^2={\mathrm{a}}^2\left(\frac{\sqrt{5}+1}{4}\right)\mathrm{ANS}... \end{gathered}$$