Question 21

Mathematics Sequence And Series Hard

If nonzero numbers \( a, b, c \) are in Harmonic Progression (H.P.), then the straight line \[ \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \] always passes through a fixed point. What is the fixed point?

(A) (1, -2)
(B) \(\left(1,\frac 1 2\right)\)
(C) (-1, 2)
(D) None
โœ“ Correct Answer: Option A Verified Solution

๐Ÿ“˜ Step-by-Step Solution & Detailed Explanation:

The correct option is A (1, -2).

Given that \( a, b, c \) are in Harmonic Progression (H.P).
Therefore, \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P).
The condition for A.P is:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
Rearranging the terms to match the form of a linear equation:
\[ \frac{1}{a} - \frac{2}{b} + \frac{1}{c} = 0 \quad \text{...(i)} \]
The given equation of the line is:
\[ \frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0 \quad \text{...(ii)} \]
Comparing equation (i) and (ii), we observe that the variable coefficients match if:
\[ x = 1 \quad \text{and} \quad y = -2 \]
Thus, the line always passes through the fixed point \( (1, -2) \).

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