Answer:
Option A,D
Explanation:
Given, (x2+xy+4x+2y+4)dydx−y2=0
⇒ [(x2+4x+4)+y(x+2)]dydx−y2=0
⇒ [(x+2)2+y(x+2)]dydx−y2=0
Put x+2=X and y=Y, then
(X2+XY)dYdX−Y2=0
⇒ X2dY+XYdY−Y2dX=0
⇒ X2dY+Y(XdY−YdX)=0
⇒ −dYY=XdY−YdXX2
⇒ −d(log|Y|)=d(YX)
On integrating both sides , we get,
−log|Y|=YX+C
where x+2=X and y=Y
⇒ −log|y|=Yx+2+C ...........(i)
Since, it passes through the point (1,3)
∴ −log3=1+C
⇒ C=−1−log3
=−(loge+log3)
=−log3e
∴ Eq. (i) becomes
log|y|+yx+2−log(3e)=0
⇒ log(|y|3e)+yx+2=0 ........(ii)
Now, to check option (a) y=x+2 intersects the curve.
⇒ log(|x+2|3e)+x+2x+2=0
⇒ log(|x+2|3e)=−1
|x+2|3e=e−1=1e
⇒ |x+2|=3 or x+2=±3
∴ x=1, -5 (rejected) , as x>0 [given]
∴ x=1 only one solution
Thus ,(a) is the correct answer.
To check option (c) we have
y=(x+2)2
and log(|y|3e)+yx+2=0
⇒ log[|x+2|23e]+(x+2)2x+2=0
⇒ log[|x+2|23e]=−(x+2)
⇒ (x+2)23e=e−(x+2)
or (x+2)2.ex+2=3e
⇒ ex+2=3e(x+2)2

Clearly, they have no solution.
To check option (d),
y=(x+3)2
i.e, log[|x+3|23e]+(x+3)2(x+2)=0
To check the number of solutions,
Let g(x)=2log(x+3)+(x+3)2(x+2)−log(3e)
∴ g′(x)=2x+3+
((x+2).2(x+3)−(x+3)2.1(x+2)2)−0
=2x+3+(x+3)(x+1)(x+2)2
Clearly, when x>0, then ,g' (x)>0
∴ g(x) is increasing , when x>0
Thus ,when x.0, then g(x) >g(0)
g(x)>log(3e)+94>0
Hence, there is no solution
Thus, option (d) is true