SATHEE: Theory of Equations 1 Question 44

Theory of Equations 1 Question 44

45. If $x^{2} - 10 a x - 11 b = 0$ have roots c and d. $x^{2} - 10 c x - 11 d = 0$ have roots $a$ and $b$ , then find $a + b + c + d$ .

$(2006, 6 M)$

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Answer:

Correct Answer: 45. (1210)

Solution:

Here,

$a + b = 10 c and c + d = 10 a$

$\begin{aligned} \Rightarrow & (a - c) + (b - d) & = 10 (c - a) \Rightarrow & (b - d) & = 11 (c - a) \end{aligned}$

Since, ’ $c$ ’ is the root of $x^{2} - 10 a x - 11 b = 0$

$c^{2} - 10 a c - 11 b = 0$

Similarly, ’ $a$ ’ is the root of

$\begin{aligned} x^{2} - 10 c x - 11 d & = 0 \\ \Rightarrow a^{2} - 10 c a - 11 d & = 0 \end{aligned}$

On subtracting Eq. (iv) from Eq. (ii), we get

$(c^{2} - a^{2}) = 11 (b - d)$

$∴ (c + a) (c - a) = 11 \times 11 (c - a)$

$\Rightarrow c + a = 121$

$∴ a + b + c + d = 10 c + 10 a$

$= 10 (c + a) = 1210$

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Theory of Equations 1 Question 44

45. If x2−10ax−11b=0 have roots c and d. x2−10cx−11d=0 have roots a and b, then find a+b+c+d.

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45. If $x^{2} - 10 a x - 11 b = 0$ have roots c and d. $x^{2} - 10 c x - 11 d = 0$ have roots $a$ and $b$ , then find $a + b + c + d$ .