Question

If a circle passes through the points \((2,3)\) and \((4,5)\) and its center lies on the straight line \(y-4x+3 = 0\), then its equation is......

• A. \(x^2+y^2-4x-10y+25 = 0\)
• B. \(x^2+y^2-4x-10y-25 = 0\)
• C. \(x^2+y^2-4x+10y-25 = 0\)
• D. \(x^2+y^2+25 = 0\)

Hint:

Centre lies on the perpendicular bisector of the chord and on the given line.

Answer 1

The Correct Option is A

Step 1: Use the general form
$x^2+y^2+2gx+2fy+c=0$ with centre $(-g,-f)$ on $y-4x+3=0$: $-f+4g+3=0$.

Step 2: Use the two points
$(2,3)$: $4g+6f+c=-13$. $(4,5)$: $8g+10f+c=-41$. Subtract: $4g+4f=-28$, so $g+f=-7$. With $f=4g+3$: $5g=-10$, $g=-2$, $f=-5$, $c=-13+8+30=25$. Equation: $x^2+y^2-4x-10y+25=0$, option (A).

Final Answer:
The circle equation is that of option (A). \[ \boxed{x^2+y^2-4x-10y+25=0} \]