The correct option is D the length of the latus rectum of the parabola is 4.
Let S1:x2+y2+6x−24y+72=0
S2:x2−y2+6x+16y−46=0
The family of curves will be,
S1+λS2=0⇒(1+λ)x2+(1−λ)y2+6(1+λ)x +(16λ−24)y+72−46λ=0
Comparing it to general two degree curve,
ax2+by2+2hxy+2fx+2gy+c=0
For the equation to represent parabola,
h2=ab
⇒(1+λ)(1−λ)=0⇒λ=±1
When λ=−1
S1−S2=0⇒y2−20y+59=0
This is not a parabola.
When λ=1
S1+S2=0⇒x2+6x−4y+13=0⇒(x+3)2=4(y−1)
The focus is (−3,1+1)=(−3,2)
Directrix is y=1−1=0
Length of latus rectum is 4