Question:
A circle passes through \((-2, 4)\) and touches the \(y\)-axis at \((0, 2)\). Which one of the following equations can represent a diameter of this circle ?
A circle passes through \((-2, 4)\) and touches the \(y\)-axis at \((0, 2)\). Which one of the following equations can represent a diameter of this circle ?
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Find the centre first (on \(y=2\), equidistant from the given point and the tangent line) and then see which line passes through it.
Updated On: Nov 11, 2026
- \(4x+5y-6=0\)
- \(2x-3y+10=0\)
- \(3x+4y-3=0\)
- \(5x+2y+4=0\)
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The Correct Option is B
Solution and Explanation
The circle touches the y-axis at \((0,2)\), so its centre lies on the horizontal line \(y=2\): centre \((h,2)\), radius \(|h|\).
It passes through \((-2,4)\): \((-2-h)^2+(4-2)^2=h^2\Rightarrow 4+4h+h^2+4=h^2\Rightarrow h=-2\).
Centre \(=(-2,2)\), radius 2.
A diameter is a line through the centre. Test: (1) \(-8+10-6=-4\ne0\); (2) \(-4-6+10=0\), yes; (3) \(-6+8-3=-1\ne0\); (4) \(-10+4+4=-2\ne0\).
Answer: (2) \(2x-3y+10=0\)
It passes through \((-2,4)\): \((-2-h)^2+(4-2)^2=h^2\Rightarrow 4+4h+h^2+4=h^2\Rightarrow h=-2\).
Centre \(=(-2,2)\), radius 2.
A diameter is a line through the centre. Test: (1) \(-8+10-6=-4\ne0\); (2) \(-4-6+10=0\), yes; (3) \(-6+8-3=-1\ne0\); (4) \(-10+4+4=-2\ne0\).
Answer: (2) \(2x-3y+10=0\)
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