If M is a foot of the perpendicular y²=4ax to its directrix and SPMis an equilateral triangle('S' is a focus),then SP=???(a>0)..

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Please find below the solution to the asked query:



$$\begin{gathered} \mathrm{Since} \mathrm{SPM}\hspace{0.17em}\mathrm{is} \mathrm{an} \mathrm{equilateral} \mathrm{triangle}: \\ \angle \mathrm{SMP}=60° \\ \mathrm{so}, \\ \angle \mathrm{SMZ}=30° \\ \mathrm{From} \mathrm{triangle} \mathrm{SZM}, \\ \frac{\mathrm{SZ}}{\mathrm{SM}}=\sin 30° \\ \frac{\mathrm{SZ}}{\mathrm{SM}}=\frac{1}{2} \\ \mathrm{SM}=2\mathrm{SZ} \\ \mathrm{we} \mathrm{know} \mathrm{SZ}=\mathrm{distance} \mathrm{between} \mathrm{focus} \mathrm{and} \mathrm{directrix}=2\mathrm{a} \\ \mathrm{so}, \mathrm{SM}=4\mathrm{a} \\ \mathbf{since}\mathbf{ }\mathbf{it}\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{an}\mathbf{ }\mathbf{equilateral}\mathbf{ }\mathbf{triangle}\mathbf{ }\mathbf{SM}\mathbf{=}\mathbf{SP}\mathbf{=}\mathbf{4}\mathbf{a} \end{gathered}$$

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