Circles
Angle Subtended by Chord
Grade 11
Question:
<p>Let A and B are two points both lying within a given circle S, and P be a point on circumference of S at which AB subtends the greatest angle.</p><p><strong>Statement-1:</strong> If \(A \equiv (1, 1)\), \(B \equiv (1, -1)\) and equation of S is \(x^2 + y^2 = 4\) then P will be \((2, 0)\)</p><p><strong>Statement-2:</strong> P will be the point where a circle passing through A and B touches the circle S.</p>
<p>(A) Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1</p>
<p>(B) Statement-1 is true, Statement-2 is true and Statement-2 is not correct explanation for Statement-1</p>
<p>(C) Statement-1 is true, Statement-2 is false</p>
<p>(D) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: The angle \(\angle APB\) is maximized when the circle through A, B, P has the smallest radius, which occurs when it is tangent to S.
<p>For a fixed chord AB inside circle S, the angle subtended is maximum at the point P on S where a circle through A and B is tangent to S. For A = (1,1) and B = (1,-1), the chord AB is vertical with midpoint (1,0). The circle through A and B with smallest radius (maximizing the angle at P) is the one tangent to S from inside. By symmetry and calculation, this tangent point is \((2, 0)\). Statement-2 is the correct geometric principle explaining Statement-1.</p>
Correct Answer: A