Question

If the sides a, b, c of ΔABC satisfy the equation 4x³ – 24x² + 47x – 30 = 0 and `|(a^2, (s - a)^2, (s - a)^2),((s - b)^2, b^2, (s - b)^2),((s - c)^2, (s - c)^2, c^2)| = p^2/q` where p and q are co-prime and s is semiperimeter of ΔABC, then the value of (p – q) is ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

If the sides a, b, c of ΔABC satisfy the equation 4x³ – 24x² + 47x – 30 = 0 and `|(a^2, (s - a)^2, (s - a)^2),((s - b)^2, b^2, (s - b)^2),((s - c)^2, (s - c)^2, c^2)| = p^2/q` where p and q are co-prime and s is semiperimeter of ΔABC, then the value of (p – q) is 7.00.

Explanation:

Given equation

4x³ – 24x² + 47x – 30 = 0

(x – 2)(2x – 5)(2x – 3) = 0

So, a = 2, b = `5/2`, c = `3/2`

s = `(a + b + c)/2` = 3

Now substituting values of a, b, c in the determined D,

D = `|(4, 1, 1),(1/4, 25/4, 1/4),(9/4, 9/4, 9/9)|` = `81/2` = `9^2/2`

∴ p – q = (9 – 2) = 7.00