Complex Numbers
Locus of Complex Numbers
Grade 11
Question:
<p><strong>Statement-1:</strong> Locus of \(z\) satisfying the equation \(|z - 1| + |z - 8| = 5\) is an ellipse.</p><p><strong>Statement-2:</strong> Sum of focal distances of any point on ellipse is constant for an ellipse.</p>
<p>(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(b) Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1</p>
<p>(c) Statement-1 is true, Statement-2 is false</p>
<p>(d) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: For an ellipse with foci $F_1$ and $F_2$, we need $|PF_1| + |PF_2| = 2a > |F_1F_2|$. Here the condition is not satisfied.
<p>For the locus $|z - 1| + |z - 8| = 5$, the sum of distances from two fixed points (1 and 8) is 5. The distance between the two foci is $|8 - 1| = 7$. Since 5 < 7, this does not form an ellipse. An ellipse requires the sum of focal distances to be greater than the distance between foci. Hence Statement-1 is false.</p><p>Statement-2 is true as it correctly defines the property of an ellipse.</p><p>∴ Answer is (d).</p>
Correct Answer: D