If a and b are rational and b is not a perfect square, then the quadratic equation with rational coefficient whose one root is (1/a+√b) is

Question

If a and b are rational and b is not a perfect square, then the quadratic equation with rational coefficient whose one root is (1/a+√b) is (A) x2-2 a x+(a2-b)=0 (B) (a2-. b) x2-2 a x+1=0 (C) (a2-b2) x2-2 b x+1=0 (D) x2+(a2-b2) x+(a2+b2)=0

Tardigrade Admin · Accepted Answer

Clearly other root is (1/a-√b) Hence sum of roots =(1/a+√b)+(1/a-√b)=(2 a/a2-b) and product of roots =(1/a2-b) so required quadratic equation is x2-(2 a/a2-b) x+(1/a2-b)=0 ⇒(a2-b) x2-2 a x+1=0

Q. If $a$ and $b$ are rational and $b$ is not a perfect square, then the quadratic equation with rational coefficient whose one root is $\frac{1}{a + b}$ is

$x^{2} - 2 a x + (a^{2} - b) = 0$

$(a^{2} -$ b) $x^{2} - 2 a x + 1 = 0$

$(a^{2} - b^{2}) x^{2} - 2 b x + 1 = 0$

$x^{2} + (a^{2} - b^{2}) x + (a^{2} + b^{2}) = 0$

Q. If a and b are rational and b is not a perfect square, then the quadratic equation with rational coefficient whose one root is a+b​1​ is

x2−2ax+(a2−b)=0

(a2− b) x2−2ax+1=0

(a2−b2)x2−2bx+1=0

x2+(a2−b2)x+(a2+b2)=0

Clearly other root is a−b​1​ Hence sum of roots =a+b​1​+a−b​1​=a2−b2a​ and product of roots =a2−b1​ so required quadratic equation is x2−a2−b2a​x+a2−b1​=0 ⇒(a2−b)x2−2ax+1=0

Q. If $a$ and $b$ are rational and $b$ is not a perfect square, then the quadratic equation with rational coefficient whose one root is $\frac{1}{a + b}$ is

$x^{2} - 2 a x + (a^{2} - b) = 0$

$(a^{2} -$ b) $x^{2} - 2 a x + 1 = 0$

$(a^{2} - b^{2}) x^{2} - 2 b x + 1 = 0$

$x^{2} + (a^{2} - b^{2}) x + (a^{2} + b^{2}) = 0$