Question

Let all the points on the curve $x^2+y^2-6x-7=0$ are reflected about the line $y=x+4$. If the locus of the reflected points is in the form $x^2+y^2+2gx+2fy+c=0$, then the value of $(g+f+c)$ is equal to:

• A. 46
• B. 40
• C. 50
• D. 56

Answer 1

Answer: (A)

46

Answer 2

The original curve is a circle $(x-3)^2+y^2=16$ with center $(3,0)$ and radius $4$. Reflection preserves the radius. The transformation for reflection about $y=x+4$ is $x=y'-4, y=x'+4$. Substituting these into the original circle equation yields the locus $x'^2+y'^2+8x'-14y'+49=0$. Comparing this to $x^2+y^2+2gx+2fy+c=0$, we find $g=4, f=-7, c=49$. The sum $g+f+c = 4-7+49=46$.