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  • If the normal at the points  <img alt="\mathrm{P_i\left(x_i, y_i\right), i=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BP_i%5Cleft%28x_i%2C%20y_i%5Cright%29%2C%20i

If the normal at the points  \mathrm{P_i\left(x_i, y_i\right), i=1}  to 4 on the hyperbola\mathrm{ x y=c^2}are concurrent at the point  \mathrm{Q}(\mathrm{h}, \mathrm{k}),  then \frac{\left(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4\right)\left(\mathrm{y}_1+\mathrm{y}_2+\mathrm{y}_3+\mathrm{y}_4\right)}{\mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4}    is equal to

Option: 1

\frac{\mathrm{hk}}{\mathrm{c}^4}


Option: 2

\frac{\mathrm{h}^2 \mathrm{k}^2}{\mathrm{c}^6}


Option: 3

\frac{\sqrt{|h \mathrm{k}|}}{\mathrm{c}^3}


Option: 4

-\frac{\mathrm{hk}}{\mathrm{c}^4}


Answers (1)

best_answer

Equation of normal at any point  \mathrm{P}\left(\mathrm{ct}, \frac{\mathrm{c}}{\mathrm{t}}\right)

on  x y=c^2 , is \mathrm{xt}^3-\mathrm{yt}-\mathrm{ct}+\mathrm{c}=0

If it passes through  \mathrm{Q}(\mathrm{h}, \mathrm{k}) then  \mathrm{ct}^4-\mathrm{ht} \mathrm{t}^3+\mathrm{kt}-\mathrm{c}=0

If it's roots are t_1, t_2, t_3 and  t_4, then
\mathrm{t}_1+\mathrm{t}_2+\mathrm{t}_3+\mathrm{t}_4=\mathrm{h} / \mathrm{c} \Rightarrow \mathrm{ct}_1+\mathrm{ct}_2+\mathrm{ct}_3+\mathrm{ct}_4 \\
=\mathrm{h} \\
\Rightarrow \Sigma \mathrm{x}_{\mathrm{i}}=\mathrm{h}, \Sigma \mathrm{t}_1 \mathrm{t}_2 \mathrm{t}_3=-\frac{\mathrm{k}}{\mathrm{c}}, \mathrm{t}_1 \mathrm{t}_2 \mathrm{t}_3 \mathrm{t}_4=-1 \\

\Rightarrow\left(\mathrm{ct}_1\right)\left(\mathrm{ct}_2\right)\left(\mathrm{ct}_3\right)\left(\mathrm{ct}_4\right)=-\mathrm{c}^4 \\

\Rightarrow \Sigma \frac{\mathrm{c}}{\mathrm{t}_{\mathrm{i}}}=\mathrm{k} \Rightarrow \Sigma \mathrm{y}_1=\mathrm{k} and    \mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4=- \\


\mathrm{c}^4 \Rightarrow \frac{\sum \mathrm{x}_{\mathrm{i}} \Sigma \mathrm{y}_{\mathrm{i}}}{\mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4}=-\frac{\mathrm{hk}}{\mathrm{c}^4}

Posted by

rishi.raj

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