Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Locus of the middle points of normal chords of the parabola y2 = 4x isOption: 1 <img alt="y^{4}+2 (x-2 ) y^{2}+16 =0" src="

Locus of the middle points of normal chords of the parabola y2 = 4x is

Option: 1

y^{4}+2 (x-2 ) y^{2}+16 =0


Option: 2

y^{4}-2 (x-2 ) y^{2}+16 =0


Option: 3

y^{4}+2 (x-2 ) y^{2}+8 =0


Option: 4

y^{4}-2 (x-2 ) y^{2}+8 =0


Answers (1)

best_answer

Equation of normal chord at any point (at2, 2at) of the parabola y2 = 4ax is

y + tx = 2at + at3.

So the equation of the normal chord is

y + tx =2t + t3.                                                                       ...(i)

M(x1, y1) is mid point of normal chord, its equation must be

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}T=S_1\\\Rightarrow \;\;\;\;\;\;\;\;\;y y_{1}-2 \left(x+x_{1}\right)=y_{1}^{2}-4 x_{1}\\\Rightarrow \;\;\;\;\;\;\;\;\;\;\;\;\;\;y y_{1}-2 x=y_{1}^{2}-2 x_{1}\;\;\;\;\;\;\;\;\;\;\;\;\ldots(ii) 

Equation (i) and (ii) are identical, compare equation (i) and (ii)

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\frac{1}{y_{1}}=\frac{t}{-2 }=\frac{2 t+ t^{3}}{y_{1}^{2}-2 x_{1}}\\\text{From first two relations }\;\;\;\;t=-\frac{2}{y_1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(iii)\\\text{From last two relations}\\\mathrm{\;\;\;\;}\;\;\;\;\;\frac{t}{-2 }=\frac{2 t+ t^{3}}{y_{1}^{2}-2 x_{1}}\\ {\Rightarrow \quad \frac{y_{1}^{2}-2 x_{1}}{-2 }=2 + t^{2}} \\ {\Rightarrow \frac{y_{1}^{2}-2 x_{1}}{-2 }=2 +\left(\frac{-2 }{y_{1}}\right)^{2}}\;\;\;\;\;\;\;\;\;\text{from eq (iii)}\\

\Rightarrow \;\;\;\frac{y_{1}^{2}-2 x_{1}}{-2 }=\frac{2 y_{1}^{2}+4 }{y_{1}^{2}}

\Rightarrow \;\;\;y^{4}-2 (x-2 ) y^{2}+8 =0

 

Posted by

Shailly goel

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 2025: NIT Rourkela cut-offs for CSE, chemical engineering drop; last years' closing ranks
anam-khan

JEE Main 2025 Live: NTA exam city slip at jeemain.nta.nic.in soon; session 1 schedule
anam-khan

JEE Main 2025 schedule out for session 1; paper 1 from January 22

Latest Question