A) 3
B) 4
C) 5
D) 7
E) 9
Correct Answer: B
Solution :
The equation of curves are \[{{x}^{2}}=9A(9-y)\] ....(i) and \[{{x}^{2}}=A(y+1)\] ...(ii) On differentiating Eq. (i), we get \[2x=-9A\frac{dy}{dx}\] \[\Rightarrow \] \[\frac{dy}{dx}=-\frac{2x}{9A}\] ???(iii) And on differentiating Eq. (ii), we get \[2x=A\frac{dy}{dx}\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{2x}{A}\] ?. (iv) Since, these curves (i) and (ii) intersect orthogonally. \[\therefore \] \[\left( -\frac{2x}{9A} \right)\left( \frac{2x}{A} \right)=-1\] From Eqs. (i) and (ii), we get \[9A(9-y)=A(y+1)\] \[\Rightarrow \] \[81-9y=y+1\] \[\Rightarrow \] \[10y=80\] \[\Rightarrow \] \[y=8\] \[\therefore \]From Eq. (i), we get \[{{x}^{2}}=9A(9-8)=9A\] From Eq.(v), \[\frac{4{{x}^{2}}}{9{{A}^{2}}}=1\] \[\Rightarrow \] \[\frac{4.9A}{9{{A}^{2}}}=1\Rightarrow A=4\]
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