Question

Equation of the chord of the circle $x^2 + y^2 - 4x - 10y + 25 = 0$ having mid-point $(1, 2)$ is

• A. $-x + 3y = 5$
• B. $x + 3y = 7$
• C. $5x + y = 7$
• D. $3x + y = 5$

Hint:

Instead of computing slopes or expanding chord formulas, check which options pass through the given midpoint $(1, 2)$! Testing option (B): $1 + 3(2) = 1 + 6 = 7$. Since it satisfies the midpoint constraint perfectly, it confirms the answer rapidly.

Answer 1

The Correct Option is B

Step 1: Identify the centre.
For $x^2+y^2-4x-10y+25=0$, the centre is $(-g,-f) = (2,5)$.
Step 2: Use the chord-midpoint geometry.
The radius to the midpoint $M(1,2)$ is perpendicular to the chord, so the chord's slope is the negative reciprocal of the slope of $CM$.
Step 3: Slope of $CM$.
$m_{CM} = \dfrac{2-5}{1-2} = \dfrac{-3}{-1} = 3$.
Step 4: Slope of the chord.
Perpendicular slope $= -\dfrac{1}{3}$.
Step 5: Point-slope equation.
Through $M(1,2)$: $y-2 = -\dfrac{1}{3}(x-1)$. Multiplying by $3$: $3y-6 = -(x-1)$.
Step 6: Tidy up.
This becomes $x + 3y = 7$, the same chord the $T=S_1$ rule produces, but here found purely from the perpendicular-radius property.
\[ \boxed{x + 3y = 7} \]