Circles
Circle
Allen Star Batch
Grade 11
Question:
$A$ and $B$ are two points in $xy$ plane, which are $2\sqrt{2}$ unit distance apart and subtend an angle of $90°$ at $C(1,2)$ on the line $x - y + 1 = 0$ which is larger than any angle subtend by the line segment $AB$ at any other point on the line. The equation of the circle through the points $A$, $B$ and $C$ is:
$x^2 + y^2 - 6x + 7 = 0$
$x^2 + y^2 - 4x + 2y + 3 = 0$
$x^2 + y^2 - 6y + 7 = 0$
$x^2 + y^2 - 4x - 2y + 3 = 0$
Step-by-Step Solution
Key Concept: When a chord subtends a right angle at the circumference, it must be a diameter; use this to find the radius.
The circle passes through $A$ and $B$ on line $x - y + 1 = 0$ and is tangent to this line at $C$. The equation is $(x-1)^2 + (y-2)^2 + \lambda(x-y+1) = 0$. Since $\angle ACB = \frac{\pi}{2}$, $AB$ is a diameter. Solving $\left(\frac{\lambda-2}{2}\right)^2 + \left(\frac{\lambda+4}{2}\right)^2 - (5+\lambda) = 2$ gives $\lambda = \pm 2$, hence radius $\sqrt{2}$.
Correct Answer: 3,4