Question

If A is the A.M of the roots of the equation \[{x^2} - 2ax +{b^2} = 0\] and G is the G.M. of the roots of the equation \[{x^2} - 2bx + {a^2} = 0\], then
A. \[A > G\]
B. \[A \ne G\]
C. \[A = G\]
D. None of these

Answer 1

Hint
The values of the variable that satisfy a quadratic equation are known as its roots. The "solutions" or "zeroes" of the quadratic equation are other names for them. The x-coordinates of the x-intercepts of a quadratic function are the roots of the function. It is a quadratic equation in its general form, where "a" stands for the leading coefficient and "c" for the absolute term of f. (x).
The roots of a quadratic equation are the values of the variables that fulfil the equation. The constant term (the third term), divided by the leading coefficient, is equivalent to the product of the roots of a quadratic equation. Future courses will teach you that these relationships also apply to equations of higher degrees.
Formula use:
To find the ratio of two numbers,
G.M of a and b \[ = \sqrt {(ab)} \]
A.M of a and b \[ =\frac{{a + b }}{2} \]
If \[{x^2} - 2ax + b = 0\] is \[\alpha \] and \[\beta \]
then sum of root, \[\alpha + \beta = 2a\]
Product of roots, \[\alpha \beta = {b^2}\]
Complete step-by-step solution
Assume that the roots of the equation \[{x^2} - 2ax + b = 0\] is \[\alpha \] and \[\beta \]
Then, the equation becomes
\[\alpha + \beta = 2a\] and
The product of roots becomes
\[\alpha \beta = {b^2}\]
Assume that \[\alpha ',\beta '\]be the roots of the equation\[{x^2} - 2bx + {a^2} = 0\]
\[ = > \alpha ' + \beta ' = 2b\] and
The product of roots becomes
\[ = > \alpha '\beta ' = {a^2}\]
Then, by equating both the equation, it becomes
\[ = > \frac{{\alpha + \beta }}{2} = \sqrt {\alpha '\beta '} \]
Hence, it is proved that \[A = G\]
Therefore, the correct option is C.

Note
The product of two binomials can be used to represent a quadratic equation. The roots of the given quadratic equation in this case are a and b. The "solutions" or numerical values that are equal to the given equation's variable are known as the roots of the equation. The x-intercepts of a function are its roots. The y-coordinate of a point on the x-axis is always zero.