<p>The locus of the point of intersection of the tangent to the circle \(x^2 + y^2 = a^2\), which include an angle of \(45°\) is the curve \((x^2 + y^2)^2 = \lambda a^2(x^2 + y^2 - a^2)\). The value of \(\lambda\) is:</p>
Step-by-Step Solution
Key Concept: Relate the angle between tangents to the distance from the external point using trigonometric relations.
<p>Let two tangents from point \(P(x,y)\) to the circle \(x^2 + y^2 = a^2\) include an angle of \(45°\). If the distance from P to the center O is r, and the tangent touches at points making angle \(\alpha\) with OP, then \(\sin\alpha = \frac{a}{r}\). For angle between tangents to be \(45°\), we have \(2\alpha = 45°\), so \(\sin(22.5°) = \frac{a}{r}\). Using the identity and simplifying the locus equation, we find \(\lambda = 8\).</p>
Correct Answer: C