If a,b,c are positive and are in A.P., the roots of the quadratic equation `ax^(2)+bx +c=0` are real for

Options

A. `|(c )/(a)-7| ge 4 sqrt(3)`
B. `|(a)/(c ) -7| lt 4 sqrt(3)`
C. all a and c
D. no a and c

Answer

To determine the conditions under which the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real, given that \( a, b, c \) are positive and in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Understand the condition for A.P.
Since \( a, b, c \) are in A.P., we have the relationship:
\[ 2b = a + c \]
This can be rearranged to express \( b \):
\[ b = \frac{a + c}{2} \]

### Step 2: Use the discriminant condition for real roots.
The roots of the quadratic equation are real if the discriminant \( D \) is non-negative:
\[ D = b^2 - 4ac \geq 0 \]

### Step 3: Substitute \( b \) into the discriminant.
Substituting \( b = \frac{a + c}{2} \) into the discriminant:
\[ D = \left(\frac{a + c}{2}\right)^2 - 4ac \]

### Step 4: Expand the expression.
Expanding the squared term:
\[ D = \frac{(a + c)^2}{4} - 4ac \]
\[ D = \frac{a^2 + 2ac + c^2}{4} - 4ac \]

### Step 5: Combine the terms.
To combine the terms, we need a common denominator:
\[ D = \frac{a^2 + 2ac + c^2 - 16ac}{4} \]
\[ D = \frac{a^2 + c^2 - 14ac}{4} \]

### Step 6: Set the discriminant greater than or equal to zero.
For the roots to be real, we need:
\[ a^2 + c^2 - 14ac \geq 0 \]

### Step 7: Rearranging the inequality.
Rearranging gives:
\[ a^2 - 14ac + c^2 \geq 0 \]

### Step 8: Recognize this as a quadratic in \( a \).
This is a quadratic inequality in \( a \):
\[ a^2 - 14ac + c^2 \geq 0 \]

### Step 9: Find the discriminant of this quadratic.
The discriminant of this quadratic in \( a \) is:
\[ D' = (-14c)^2 - 4 \cdot 1 \cdot c^2 = 196c^2 - 4c^2 = 192c^2 \]

### Step 10: Analyze the roots.
Since \( D' \) is positive, this quadratic has two distinct real roots. The quadratic opens upwards (as the coefficient of \( a^2 \) is positive), so it will be non-negative outside the interval defined by its roots.

### Step 11: Find the roots.
The roots can be found using the quadratic formula:
\[ a = \frac{14c \pm \sqrt{192c^2}}{2} \]
\[ = 7c \pm 4\sqrt{3}c \]
Thus, the roots are:
\[ a_1 = (7 + 4\sqrt{3})c, \quad a_2 = (7 - 4\sqrt{3})c \]

### Step 12: Conclusion.
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real if \( a \) lies outside the interval:
\[ a \leq (7 - 4\sqrt{3})c \quad \text{or} \quad a \geq (7 + 4\sqrt{3})c \]