Question

Let all the points on the curve \[ x^{2}+y^{2}-10x=0 \] are reflected about the line \(y=x+3\). If the locus of the reflected points is in the form \[ x^{2}+y^{2}+gx+fy+c=0, \] then the value of \((g+f+c)\) is:

• A. 38
• B. -28
• C. 28
• D. -38

Hint:

Reflections are rigid transformations, meaning they never change the size or shape of a geometric figure. For circles, you can always skip transforming the entire equation—just move the center point like a regular dot and keep the radius value exactly the same!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Reflecting a circle about a line results in another circle with the same radius but a different center. The new center is the reflection of the original center across the given line.
Step 2: Key Formula or Approach:
1. Find the center and radius of the original circle.
2. Reflect the center \( (h, k) \) about the line \( ax + by + c = 0 \) using the formula: \[ \frac{x-h}{a} = \frac{y-k}{b} = -2 \frac{ah + bk + c}{a^2 + b^2} \] 3. Write the equation of the new circle.
Step 3: Detailed Explanation:
Original circle: \( x^2 + y^2 - 10x = 0 \implies (x-5)^2 + y^2 = 25 \).
Center \( C_1 = (5, 0) \), Radius \( R = 5 \).
Line: \( x - y + 3 = 0 \).
Reflect \( (5, 0) \):
\[ \frac{x-5}{1} = \frac{y-0}{-1} = -2 \frac{(1)(5) - (1)(0) + 3}{1^2 + (-1)^2} = -2 \frac{8}{2} = -8 \]
\( x-5 = -8 \implies x = -3 \).
\( y/(-1) = -8 \implies y = 8 \).
Reflected center \( C_2 = (-3, 8) \).
New circle equation: \( (x+3)^2 + (y-8)^2 = 25 \).
Expand: \( x^2 + 6x + 9 + y^2 - 16y + 64 = 25 \).
\( x^2 + y^2 + 6x - 16y + 48 = 0 \).
Comparing with \( x^2 + y^2 + gx + fy + c = 0 \):
\( g = 6, f = -16, c = 48 \).
Value of \( g + f + c = 6 - 16 + 48 = 38 \).
Step 4: Final Answer:
The sum \( g+f+c \) is 38.