Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Length of the shortest normal chord of the parabola is<br

Length of the shortest normal chord of the parabola \mathrm{y}^2=4 \mathrm{ax} is
 

Option: 1

\mathrm{a} \sqrt{27}
 


Option: 2

3 \mathrm{a} \sqrt{3}
 


Option: 3

2 \mathrm{a} \sqrt{27}

 


Option: 4

\text{None of these}


Answers (1)

best_answer

Let \mathrm{AB} be a normal chord where  \mathrm{\mathrm{A} \equiv\left(\mathrm{at}_1{ }^2, 2 \mathrm{at}_1\right), \mathrm{B} \equiv\left(\mathrm{at}_2{ }_2, 2 \mathrm{at}_2\right). } We have \mathrm{ \mathrm{t}_2=-\mathrm{t}_1-\frac{2}{\mathrm{t}_1}. }

\mathrm{ \mathrm{AB}^2=\left[\mathrm{a}^2\left(\mathrm{t}_1-\mathrm{t}_2\right)^2\left(\mathrm{t}_1+\mathrm{t}_2\right)^2+4\right]=\mathrm{a}^2\left(\mathrm{t}_1 \mathrm{t}_1+\frac{2}{\mathrm{t}_1}\right)^2\left(\frac{4}{\mathrm{t}_1^2}+4\right)=\frac{16 \mathrm{a}^2\left(1+\mathrm{t}_1^2\right)^3}{\mathrm{t}_1^4} }

\mathrm{ \Rightarrow \quad \frac{\mathrm{d}\left(\mathrm{AB}^2\right)}{\mathrm{dt}_1}=16 \mathrm{a}^2\left(\frac{\mathrm{t}_1^4\left[3\left(1+\mathrm{t}_1^2\right)^2 \cdot 2 \mathrm{t}_1\right]-\left(1+\mathrm{t}_1^2\right)^3 \cdot 4 \mathrm{t}_1^3}{\mathrm{t}_1^8}\right)=\frac{\mathrm{a}^2 \cdot 32\left(1+\mathrm{t}_1^2\right)^2}{\mathrm{t}_1^5}\left(\mathrm{t}_1^2-2\right) }
\mathrm{\mathrm{t}_1=\sqrt{2}} is indeed the point of minima of \mathrm{\mathrm{AB}^2.}

Thus \mathrm{\mathrm{AB}_{\text {mini }}=\frac{4 \mathrm{a}}{2}(1+2)^{3 / 2}=2 \mathrm{a} \sqrt{27}\: units.}

Hence option 3 is correct.

Posted by

seema garhwal

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026 Registration LIVE: NTA JEE form at jeemain.nta.nic.in; what's new this year?
anam-khan

JEE Main 2026: Wider gap in NIT cut-offs for home and other-state quotas; outsiders need better rank
anam-khan

JEE Mains 2026: Use of calculator 'not allowed’ says NTA; cites ‘typographical error' in information bulletin

Latest Question