The following question is taken from JEE practice set :
Consider the curves
$C_1: x^2y^2-a^2(x^2+y^2)=0$ ,
$C_2: xy=a^2$ ,
$C_3: x^2+y^2=a^2(a≠0).$
Let number of points common to all three curves is $m$, total points common to any two of the curves is $n$.
Number of tangents (which are not asymptotes) common to all three curves is $p$ and number of tangents (not asymptotes) common to any two of the curves is $q$ then find $m+n, m+n+p+q$.
My attempt :
I put $C_2$ and $C_3$ in $C_1$ and got $0=0$, that means three curves do have at least one point in common?
Putting $C_2$ in $C_1$, I am getting $a^4-a^2(x^2+\frac{a^4}{x^2}$=0, that means $a^2x^2-x^4-a^4=0\implies x^4+a^4+2a^2x^2=3a^2x^2\implies x^2+a^2=±√3ax$
Its discriminant is coming out to be zero, does that mean there is no common point between $C_1$ and $C_2$?
Common Points Part :
We have $C_2:xy=a^2$ , hence $x^2y^2=a^4$
Plug that in $C_1:x^2y^2-a^2(x^2+y^2)=0$ to get $a^4-a^2(x^2+y^2)=0$ , hence $a^2=(x^2+y^2)$
Subtract $2C_2$ to get :
$(x-y)^2=(x^2+y^2)-2xy=a^2-2a^2=-a^2$
Positive number Equals Negative number : Hence , no Points in Common & $n=0$.
We can not have larger $m$ , hence $m=0$
Likewise , there are no Common Points between C2 & C3 & between C3 & C1.
Over-all , $n=0$ & $m=0$.
Common Tangents Part :
INCOMPLETE METHOD 1 :
Consider $C_2 : xy=a^2$ which gives $y=a^2/x$
Differentiate it to get $dy/dx=-a^2/x^2$.
The tangent line $y=mx+c$ will go though $(x_0,y_0)$ with $m=-a^2/x_0^2$ & $y_0=a^2/x_0$
Hence , $y_0=a^2/x_0=-a^2x_0/x_0^2+c=-a^2/x_0+c$
Thus , $c=2a^2/x_0$
The tangent lines $T_1$ are $y=-a^2x^2/x_0^2+2a^2/x_0$ for each $x_0$
Consider $C_3 : x^2+y^2=a^2$ which gives $y=\sqrt{a^2-x^2}$
Differentiate it to get $dy/dx=[1/2\sqrt{a^2-x^2}]2x$.
The tangent line $y=mx+c$ will go though $(x_0,y_0)$ with $m=[1/2\sqrt{a^2-x^2}]2x$ & $y_0=\sqrt{a^2-x_0^2}$
Hence , $y_0=\sqrt{a^2-x_0^2}=[1/2\sqrt{a^2-x_0^2}]2x_0^2+c$
Thus , $c=[a^2-2x_0^2]/\sqrt{a^2-x_0^2}$
The tangent lines $T_2$ are $y=x\sqrt{a^2-x_0^2}+[a^2-2x_0^2]/\sqrt{a^2-x_0^2}$ for each $x_0$
We have to check $T_1$ & $T_2$ to get the Common lines , which looks very cumbersome.
We have to get the number of Common tangents , we do not have to get the Equations of those tangents. Hence let us terminate it here , to try Visual Way.
VISUAL METHOD 2 :
Let $a=1$ & make the Curves $C_2$ & $C_3$ :
Trying to include Common tangents , we get 2+2=4 :
Hence $q=4$
When we check with C1 & C2 , we get :
We have $q=2$ , though those are asymptotes , hence not contributing over-all.
When we check with C3 & C1 , we get :
We have $q=4$ , though those are asymptotes , hence not contributing over-all.
When we try with all 3 Curves , we get :
Obviously no Common tangents ! Hence $p=0$
Overall , we have $q=4+2+0=6$ & $p=0+0+0=0$.
$m+n=0$
$m+n+p+q=6$
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