If `a,b,c` are in H.P and `ab+bc+ca=15` then `ca=`

Options

A. ad
B. 2ad
C. 3ad
D. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the condition of Harmonic Progression (H.P.)
Since \( a, b, c \) are in H.P., it implies that their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.). This means:
\[ b = \frac{2ac}{a + c} \]

### Step 2: Use the given equation
We are given that:
\[ ab + bc + ca = 15 \]
We need to express this equation in terms of \( a, b, c \).

### Step 3: Substitute the value of \( b \)
Substituting \( b \) from the H.P. condition into the equation:
\[ ab + bc + ca = 15 \]
Substituting \( b = \frac{2ac}{a+c} \):
\[ a \left(\frac{2ac}{a+c}\right) + \left(\frac{2ac}{a+c}\right)c + ca = 15 \]

### Step 4: Simplify the equation
Now, we can simplify the left side:
\[ \frac{2a^2c}{a+c} + \frac{2ac^2}{a+c} + ca = 15 \]
Combining the first two terms:
\[ \frac{2ac(a+c)}{a+c} + ca = 15 \]
This simplifies to:
\[ 2ac + ca = 15 \]
Now, factor out \( ca \):
\[ ca(2 + 1) = 15 \]
Thus:
\[ 3ac = 15 \]

### Step 5: Solve for \( ac \)
Now, divide both sides by 3:
\[ ac = \frac{15}{3} = 5 \]

### Step 6: Find \( ca \)
Since \( ca = ac \), we conclude:
\[ ca = 5 \]

### Final Answer
Thus, the value of \( ca \) is \( 5 \).

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