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  • Let  be the set of all <img alt="\mathrm{a\in N}" src="https://entrancecorner.oncodecogs.com

Let \mathrm{S} be the set of all \mathrm{a\in N} such that the area of the triangle formed by the tangent at the point\mathrm{P(b,c),b,c\in \mathbb{N}}, on the parabola \mathrm{y^{2}=2ax} and the lines \mathrm{x = b, y= 0}  is \mathrm{16\: unit^{2}}, then \mathrm{\sum_{a\in s}^{}}    \mathrm{a} is equal to
 

Option: 1

146


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

tangent at \mathrm{P(b, c) my^{2} = 2ax} is

\mathrm{ \text { area }=\left|\frac{1}{2} \times 2 \mathrm{~b} \times \frac{2 \mathrm{ba}}{\mathrm{c}}\right|=16 }
\mathrm{ \frac{2 \mathrm{~b}^2 \mathrm{a}}{\mathrm{C}}=16 }
\mathrm{ \frac{\mathrm{b}^2 \mathrm{a}}{\mathrm{C}}=8 }
\mathrm{ \because \mathrm{P}(\mathrm{b}, \mathrm{c}) \text { lies on } \mathrm{y}^2=2 \mathrm{ax} }
\mathrm{ \mathrm{C}^2=2 \mathrm{ab} }
\mathrm{ \Rightarrow \quad \frac{\mathrm{b}^4 \mathrm{a}^2}{\mathrm{c}^2}=64 }
\mathrm{ \Rightarrow \quad \frac{\mathrm{b}^4 \mathrm{a}^2}{2 \mathrm{ab}}=64 }
\mathrm{ \Rightarrow \quad \mathrm{b}^3 \mathrm{a}=128 }
\mathrm{ \Rightarrow \quad a=\frac{128}{\mathrm{~b}^3} }
\mathrm{ \mathrm{a} \: can\: \mathrm{be} \: 128,16,2 \text { then } }
\mathrm{ \mathrm{S}=\{2,16,128\} }
\mathrm{ \sum \mathrm{a}=146}
 

 

Posted by

Ritika Harsh

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