Question:
The equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line \( y - 4x + 3 = 0 \), is:
The equation of the circle which passes through the points (2, 3) and (4, 5) and the centre lies on the straight line \( y - 4x + 3 = 0 \), is:
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To find the equation of a circle given two points on it and the condition that the center lies on a straight line, use the general equation and substitute the given points and center condition.
Updated On: Apr 22, 2026
- \( x^2 + y^2 + 4x - 10y + 25 = 0 \)
- \( x^2 + y^2 - 4x - 10y + 16 = 0 \)
- \( x^2 + y^2 - 4x - 10y + 25 = 0 \)
- None of these
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The Correct Option is C
Solution and Explanation
Step 1: General equation of a circle.
The general equation of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \] where the center is \( (-g, -f) \) and the radius is \( \sqrt{g^2 + f^2 - c} \).
Step 2: Use the fact that the center lies on the given line.
We are given that the center of the circle lies on the line \( y - 4x + 3 = 0 \). The center is \( (-g, -f) \), so substituting these into the line equation: \[ -f - 4(-g) + 3 = 0 \implies -f + 4g = -3 \quad \text{(Equation 1)} \]
Step 3: Use the fact that the circle passes through the points (2, 3) and (4, 5).
Substitute the point \( (2, 3) \) into the general circle equation: \[ 2^2 + 3^2 + 2g(2) + 2f(3) + c = 0 \implies 4 + 9 + 4g + 6f + c = 0 \implies 13 + 4g + 6f + c = 0 \quad \text{(Equation 2)} \] Similarly, substitute the point \( (4, 5) \): \[ 4^2 + 5^2 + 2g(4) + 2f(5) + c = 0 \implies 16 + 25 + 8g + 10f + c = 0 \implies 41 + 8g + 10f + c = 0 \quad \text{(Equation 3)} \]
Step 4: Solve the system of equations.
Now, we solve the system of equations (1), (2), and (3). Subtract Equation 2 from Equation 3: \[ (41 + 8g + 10f + c) - (13 + 4g + 6f + c) = 0 \] \[ 28 + 4g + 4f = 0 \implies 4g + 4f = -28 \implies g + f = -7 \quad \text{(Equation 4)} \] Now, substitute Equation 4 into Equation 1: \[ -f + 4g = -3 \] Substitute \( f = -7 - g \) into this equation: \[ -( -7 - g ) + 4g = -3 \implies 7 + g + 4g = -3 \implies 5g = -10 \implies g = -2 \] Substitute \( g = -2 \) into Equation 4: \[ -2 + f = -7 \implies f = -5 \]
Step 5: Final substitution to find \( c \).
Now that we have \( g = -2 \) and \( f = -5 \), substitute these values into Equation 2:
\[ 13 + 4(-2) + 6(-5) + c = 0 \implies 13 - 8 - 30 + c = 0 \implies -25 + c = 0 \implies c = 25 \]
Step 6: Equation of the circle.
Thus, the equation of the circle is:
\[ x^2 + y^2 - 4x - 10y + 25 = 0 \]
This matches option (C).
The general equation of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \] where the center is \( (-g, -f) \) and the radius is \( \sqrt{g^2 + f^2 - c} \).
Step 2: Use the fact that the center lies on the given line.
We are given that the center of the circle lies on the line \( y - 4x + 3 = 0 \). The center is \( (-g, -f) \), so substituting these into the line equation: \[ -f - 4(-g) + 3 = 0 \implies -f + 4g = -3 \quad \text{(Equation 1)} \]
Step 3: Use the fact that the circle passes through the points (2, 3) and (4, 5).
Substitute the point \( (2, 3) \) into the general circle equation: \[ 2^2 + 3^2 + 2g(2) + 2f(3) + c = 0 \implies 4 + 9 + 4g + 6f + c = 0 \implies 13 + 4g + 6f + c = 0 \quad \text{(Equation 2)} \] Similarly, substitute the point \( (4, 5) \): \[ 4^2 + 5^2 + 2g(4) + 2f(5) + c = 0 \implies 16 + 25 + 8g + 10f + c = 0 \implies 41 + 8g + 10f + c = 0 \quad \text{(Equation 3)} \]
Step 4: Solve the system of equations.
Now, we solve the system of equations (1), (2), and (3). Subtract Equation 2 from Equation 3: \[ (41 + 8g + 10f + c) - (13 + 4g + 6f + c) = 0 \] \[ 28 + 4g + 4f = 0 \implies 4g + 4f = -28 \implies g + f = -7 \quad \text{(Equation 4)} \] Now, substitute Equation 4 into Equation 1: \[ -f + 4g = -3 \] Substitute \( f = -7 - g \) into this equation: \[ -( -7 - g ) + 4g = -3 \implies 7 + g + 4g = -3 \implies 5g = -10 \implies g = -2 \] Substitute \( g = -2 \) into Equation 4: \[ -2 + f = -7 \implies f = -5 \]
Step 5: Final substitution to find \( c \).
Now that we have \( g = -2 \) and \( f = -5 \), substitute these values into Equation 2:
\[ 13 + 4(-2) + 6(-5) + c = 0 \implies 13 - 8 - 30 + c = 0 \implies -25 + c = 0 \implies c = 25 \]
Step 6: Equation of the circle.
Thus, the equation of the circle is:
\[ x^2 + y^2 - 4x - 10y + 25 = 0 \]
This matches option (C).
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