Ellipse
Auxiliary Circle and Focal Properties
Grade 11
Question:
<p>For a point \(P\) on an ellipse, the circles with diameters \(PS\) and \(PS'\) (where \(S\), \(S'\) are the foci) intersect the auxiliary circle of the ellipse at \(A\), \(A_1\) and \(B\), \(B_1\) respectively. Which of the following is/are correct?</p>
<p>(a) \(A\) and \(A_1\) coincide, \(B\) and \(B_1\) coincide</p>
<p>(b) Segment \(AB\) is tangent to ellipse at \(P\)</p>
<p>(c) Tangents at \(A\) and \(B\) on auxiliary circles are perpendicular</p>
<p>(d) \(SA\) and \(S'B\) are parallel</p>
Step-by-Step Solution
Key Concept: Use the property that circles with focal radii as diameters have special geometric relationships with the auxiliary circle, and the resulting chord is tangent to the ellipse.
<p>For a point \(P\) on the ellipse with foci \(S\) and \(S'\):</p><p>The circle with diameter \(PS\) passes through any point \(A\) on it such that \(\angle PAS = 90°\). Similarly for \(B\) with respect to \(PS'\).</p><p>When these circles intersect the auxiliary circle (radius \(a\)), the geometry shows:</p><p>- Points \(A\) and \(B\) form the segment that is tangent to the ellipse at \(P\) (this is a classical result).</p><p>- The configuration ensures \(SA \parallel S'B\) due to the symmetry and property of the auxiliary circle.</p>
Correct Answer: b, d