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  • is latus rectum of a parabola <img alt="\mathrm{y}^{2}=4 \mathrm{ax}" src="https://entrancecorner.o

\mathrm{BC} is latus rectum of a parabola \mathrm{y}^{2}=4 \mathrm{ax} and \mathrm{A} is its vertex, then minimum length of projection \mathrm{BC} on a tangent drawn in portion \mathrm{BAC} is

Option: 1

\mathrm{a}


Option: 2

\mathrm{2 \sqrt{2} a}


Option: 3

\mathrm{2 \mathrm{a}}


Option: 4

3 \mathrm{a}


Answers (1)

best_answer

 Let tangent at \left(\mathrm{at}^{2}, 2 \mathrm{at}\right) makes an angle \theta with \mathrm{x}-axis, then \mathrm{\tan \theta=\frac{1}{\mathrm{t}}}

\Rightarrow Projection of \mathrm{BC} on tangent \mathrm{=\mathrm{BC}}

\mathrm{\sin \theta=\frac{4 \mathrm{a}}{\sqrt{1+\mathrm{t}^{2}}} \geq 2 \mathrm{a} \sqrt{2}(\, as\, -1 \leq \mathrm{t} \leq 1)}

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