Circles
Power of a Point Theorem
Grade 11
Question:
<p>Let P is situated at a distance 'd' from centre O, then which of the following does not equal the product (PQ)(PR), where T is a point on C and PT is tangent to C:</p>
<p>(a) \(d^2 - r^2\)</p>
<p>(b) \((PT)^2\)</p>
<p>(c) \((PS)^2 - (QS)(RS)\)</p>
<p>(d) \((PS)^2\)</p>
Step-by-Step Solution
Key Concept: Power of a Point theorem states that for external point P, (PQ)(PR) equals the square of tangent length and also equals d² - r².
<p><strong>Step 1:</strong> By Power of a Point theorem: \((PQ)(PR) = d^2 - r^2\)</p><p><strong>Step 2:</strong> Also, \((PT)^2 = d^2 - r^2\) since PT is tangent to circle C.</p><p><strong>Step 3:</strong> For option (c): Since S is midpoint of QR, we have \((PS)^2 - (QS)(RS) = (PS)^2 - \frac{(QR)^2}{4}\), which simplifies to \(d^2 - r^2\).</p><p><strong>Step 4:</strong> Option (d) \((PS)^2\) is generally not equal to \(d^2 - r^2\) unless special conditions hold.</p><p>∴ Answer is D.</p>
Correct Answer: d