<p>Locus of a point, whose chord of contact with respect to the circle \(x^2 + y^2 = 4\) is a tangent to the hyperbola \(xy = 1\) is a/an:</p>
Step-by-Step Solution
Key Concept: Find the chord of contact from an external point, then use the condition that this chord is tangent to the given hyperbola to establish a relationship between the coordinates of the external point.
<p><strong>Step 1: Equation of Chord of Contact</strong><br>Let P(h,k) be the external point. The chord of contact with respect to circle x²+y²=4 is:<br>hx + ky = 4</p><p><strong>Step 2: Apply Tangency Condition to Hyperbola xy=1</strong><br>This chord must be tangent to the hyperbola xy=1. Rewrite the chord as:<br>y = (4-hx)/k, where k≠0<br><br>Substituting into xy=1:<br>x·(4-hx)/k = 1<br>x(4-hx) = k<br>4x - hx² = k<br>hx² - 4x + k = 0</p><p><strong>Step 3: Apply Discriminant Condition</strong><br>For the line to be tangent to the hyperbola, this quadratic in x must have exactly one solution. Therefore, discriminant = 0:<br>Δ = 16 - 4(h)(k) = 0<br>16 - 4hk = 0<br>4hk = 16<br>hk = 4</p><p><strong>Step 4: Find the Locus</strong><br>Replacing (h,k) with (x,y), the locus of point P is:<br>xy = 4</p><p><strong>Step 5: Identify the Curve</strong><br>The equation xy = 4 is of the form xy = constant, which is a rectangular hyperbola.<br><br>∴ Answer: C</p>
Correct Answer: C