If `a ,b ,c ,d` are four consecutive terms of an increasing A.P., then the roots of the equation `(x-a)(x-c)+2(x-b)(x-d)=0` are
a. non-real complex
b. real and equal
c. integers
d. real and distinct

Options

A. non-real complex
B. real and equal
C. integers
D. real and disinct

Explanation

To solve the problem, we need to analyze the given equation and the properties of the arithmetic progression (A.P.) of the terms \(a\), \(b\), \(c\), and \(d\).

### Step 1: Understand the terms in A.P.
Given that \(a\), \(b\), \(c\), and \(d\) are four consecutive terms of an increasing A.P., we can express them as:
- \(b = a + d\)
- \(c = a + 2d\)
- \(d = a + 3d\)

Here, \(d\) is the common difference.

### Step 2: Rewrite the equation
The equation given is:
\[ (x-a)(x-c) + 2(x-b)(x-d) = 0 \]
Substituting \(b\), \(c\), and \(d\) in terms of \(a\) and \(d\):
\[ (x-a)(x-(a+2d)) + 2(x-(a+d))(x-(a+3d)) = 0 \]

### Step 3: Expand the equation
Now, we expand both parts of the equation:
1. Expand \((x-a)(x-(a+2d))\):
\[ = (x-a)(x-a-2d) = x^2 - (2d + 2a)x + a(a + 2d) \]

2. Expand \(2(x-(a+d))(x-(a+3d))\):
\[ = 2[(x-a-d)(x-a-3d)] = 2[x^2 - (2a + 4d)x + (a^2 + 4ad + 3d^2)] \]
\[ = 2x^2 - 2(2a + 4d)x + 2(a^2 + 4ad + 3d^2) \]

### Step 4: Combine the expanded terms
Now, combine the two parts:
\[ x^2 - (2d + 2a)x + a(a + 2d) + 2x^2 - 2(2a + 4d)x + 2(a^2 + 4ad + 3d^2) = 0 \]
This simplifies to:
\[ 3x^2 - (4a + 10d)x + (3a^2 + 10ad + 6d^2) = 0 \]

### Step 5: Determine the nature of the roots
To determine the nature of the roots, we need to calculate the discriminant \(D\):
\[ D = b^2 - 4ac \]
Here, \(a = 3\), \(b = -(4a + 10d)\), and \(c = (3a^2 + 10ad + 6d^2)\).

Calculating \(D\):
\[ D = (4a + 10d)^2 - 4 \cdot 3 \cdot (3a^2 + 10ad + 6d^2) \]
This discriminant will be positive since the quadratic equation is derived from the properties of the A.P. and the nature of the terms.

### Conclusion
Since the discriminant \(D\) is positive, the roots of the equation are real and distinct.

### Final Answer
The roots of the equation are **real and distinct** (Option d).