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:
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:
Show 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!
Updated On: May 28, 2026
- 38
- -28
- 28
- -38
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
Reflecting a circle across a straight line creates a new circle that preserves the exact same radius size. Therefore, we can find the equation of the reflected circle simply by finding the new coordinates of its center point using the standard line reflection transformation formula.
Step 1: Find the center and radius of the original circle.
The given circle equation is $x^2 + y^2 - 10x = 0$. Let us rewrite it in standard form by completing the square: $$(x^2 - 10x + 25) + y^2 = 25 \quad \Rightarrow \quad (x - 5)^2 + y^2 = 5^2$$ This shows that the original circle has: $$\text{Center } C_1 = (5, 0) \quad \text{and} \quad \text{Radius } R = 5$$
Step 2: Reflect the center point across the given line.
Let the coordinates of the reflected center point be $C_2(h, k)$. The formula for reflecting a point $(x_1, y_1)$ across a line $Ax + By + C = 0$ is: $$\frac{h - x_1}{A} = \frac{k - y_1}{B} = -2 \cdot \frac{Ax_1 + By_1 + C}{A^2 + B^2}$$ The given line equation is $y = x + 3 \implies x - y + 3 = 0$, where $A = 1, B = -1, C = 3$. Substituting the center point $(5, 0)$: $$\frac{h - 5}{1} = \frac{k - 0}{-1} = -2 \cdot \frac{1(5) - 1(0) + 3}{1^2 + (-1)^2}$$ $$h - 5 = -k = -2 \cdot \frac{5 + 3}{1 + 1} = -2 \cdot \frac{8}{2} = -8$$ Now solve for $h$ and $k$ independently:
• $h - 5 = -8 \implies h = -3$
• $-k = -8 \implies k = 8$ Thus, the new reflected center point is $C_2 = (-3, 8)$.
Step 3: Write out the equation of the reflected circle.
Using our new center $(-3, 8)$ and the preserved radius $R = 5$: $$(x - (-3))^2 + (y - 8)^2 = 5^2 \quad \Rightarrow \quad (x + 3)^2 + (y - 8)^2 = 25$$ Expand the polynomial expressions: $$x^2 + 6x + 9 + y^2 - 16y + 64 = 25$$ $$x^2 + y^2 + 6x - 16y + 48 = 0$$
Step 4: Extract the coefficients and calculate their sum.
Comparing this equation with the template form $x^2 + y^2 + gx + fy + c = 0$: $$g = 6, \quad f = -16, \quad c = 48$$ Calculate the requested sum of these parameters: $$\text{Sum} = g + f + c = 6 + (-16) + 48 = -10 + 48 = 38$$ This matches option (A). % Final Correct Answer Selection Update Correct Answer: (A) 38
The given circle equation is $x^2 + y^2 - 10x = 0$. Let us rewrite it in standard form by completing the square: $$(x^2 - 10x + 25) + y^2 = 25 \quad \Rightarrow \quad (x - 5)^2 + y^2 = 5^2$$ This shows that the original circle has: $$\text{Center } C_1 = (5, 0) \quad \text{and} \quad \text{Radius } R = 5$$
Step 2: Reflect the center point across the given line.
Let the coordinates of the reflected center point be $C_2(h, k)$. The formula for reflecting a point $(x_1, y_1)$ across a line $Ax + By + C = 0$ is: $$\frac{h - x_1}{A} = \frac{k - y_1}{B} = -2 \cdot \frac{Ax_1 + By_1 + C}{A^2 + B^2}$$ The given line equation is $y = x + 3 \implies x - y + 3 = 0$, where $A = 1, B = -1, C = 3$. Substituting the center point $(5, 0)$: $$\frac{h - 5}{1} = \frac{k - 0}{-1} = -2 \cdot \frac{1(5) - 1(0) + 3}{1^2 + (-1)^2}$$ $$h - 5 = -k = -2 \cdot \frac{5 + 3}{1 + 1} = -2 \cdot \frac{8}{2} = -8$$ Now solve for $h$ and $k$ independently:
• $h - 5 = -8 \implies h = -3$
• $-k = -8 \implies k = 8$ Thus, the new reflected center point is $C_2 = (-3, 8)$.
Step 3: Write out the equation of the reflected circle.
Using our new center $(-3, 8)$ and the preserved radius $R = 5$: $$(x - (-3))^2 + (y - 8)^2 = 5^2 \quad \Rightarrow \quad (x + 3)^2 + (y - 8)^2 = 25$$ Expand the polynomial expressions: $$x^2 + 6x + 9 + y^2 - 16y + 64 = 25$$ $$x^2 + y^2 + 6x - 16y + 48 = 0$$
Step 4: Extract the coefficients and calculate their sum.
Comparing this equation with the template form $x^2 + y^2 + gx + fy + c = 0$: $$g = 6, \quad f = -16, \quad c = 48$$ Calculate the requested sum of these parameters: $$\text{Sum} = g + f + c = 6 + (-16) + 48 = -10 + 48 = 38$$ This matches option (A). % Final Correct Answer Selection Update Correct Answer: (A) 38
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