<p>If \(x^2 + y^2 = r^2\) and \(z = \dfrac{1}{r}\) then \(z = \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2}\).</p><p><strong>State whether the statement is true or false.</strong></p>
Step-by-Step Solution
Key Concept: Differentiate the constraint x² + y² = r² implicitly to find dy/dx in terms of x and y, then substitute into the given expression and verify whether it actually equals z = 1/r.
<p><strong>Step 1:</strong> From the constraint x² + y² = r², differentiate implicitly with respect to x:</p><p>2x + 2y(dy/dx) = 0</p><p>∴ dy/dx = −x/y</p><p><strong>Step 2:</strong> Substitute into √(1 + (dy/dx)²):</p><p>√(1 + x²/y²) = √((y² + x²)/y²) = √(r²/y²) = r/|y|</p><p><strong>Step 3:</strong> Compare with z = 1/r:</p><p>We get r/|y|, which equals 1/r only if r²= |y|. This is NOT generally true for all points on the circle.</p><p>Since r/|y| ≠ 1/r in general, the statement is <strong>FALSE</strong>.</p><p>∴ Answer: <strong>FALSE (B)</strong></p>
Correct Answer: B