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Theory of Equations: Distinct Positive Real Numbers Roots 10cx 11d Roots

JEE Maths question with a full step-by-step solution.

Question

a, b, c, d

are distinct positive real numbers such that

a

and

b

are the roots of

x^2 - 10cx - 11d = 0

and

c

and

d

are the roots of

x^2 - 10ax - 11b = 0

, then the value of

a + b + c + d

1110

1010

1101

1210

correct

Solution

Step 1: By Vieta's formulas:

a+b = 10c, \quad ab = -11d \quad \cdots (1), \qquad c+d = 10a, \quad cd = -11b \quad \cdots (2)

Step 2: Since

a

is a root of the first equation and

c

is a root of the second

a^2 - 10ca - 11d = 0, \quad c^2 - 10ac - 11b = 0

Subtracting:

(a^2 - c^2) - 11(d - b) = 0

, and from (1):

d = 10a - c

b = 10c - a

. Step 3: Substituting and simplifying gives:

(a-c)\bigl[-(a+c) + 121\bigr] = 0

Since

a \neq c

(distinct),

a + c = 121

. Step 4:

a + b + c + d = (a + b) + (c + d) = 10c + 10a = 10(a + c) = 10 \times 121 = 1210

Correct answer: (4)

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