Question

If $\alpha $ is a root repeated twice of the quadratic equation $\left( {a - d} \right){x^2} + ax + \left( {a + d} \right) = 0$ , then $\dfrac{{{d^2}}}{{{a^2}}}$ has the value equal to
A ) ${\sin ^2}{90^o}$
B ) ${\cos ^2}{60^o}$
C ) ${\sin ^2}{45^o}$
D ) ${\cos ^2}{30^o}$

Answer 1

Hint: Use the concept of nature of roots related to the value of the discriminant in case of equal roots to find out the value of the given expression. Then use the values of standard trigonometric functions and angles to get the answer.

Complete step by step solution: It has been given that $\alpha $ is a repeated root of the quadratic equation $\left( {a - d} \right){x^2} + ax + \left( {a + d} \right) = 0$.
From this statement, it implies that both the roots are equal to each other.
We know from the nature of a quadratic equation, that if the quadratic equation$A{x^2} + Bx + C = 0$ has equal roots, it implies that the discriminant of the quadratic equation is equal to zero.
Hence, we get , $D = 0 \Rightarrow {B^2} - 4AC = 0$
In the given quadratic equation $\left( {a - d} \right){x^2} + ax + \left( {a + d} \right) = 0$, using comparing of coefficients,
$A = \left( {a - d} \right)\;\;;\;\;B = a\;\;;\;\;C = \left( {a + d} \right)$
Thus, using the discriminant formula and equating it to zero, we get
$\begin{array}{l}
{a^2} - 4\left( {a - d} \right)\left( {a + d} \right) = 0\\
 \Rightarrow {a^2} - 4\left( {{a^2} - {d^2}} \right) = 0\\
 \Rightarrow {a^2} - 4{a^2} + 4{d^2} = 0\\
 \Rightarrow 4{d^2} = 3{a^2}
\end{array}$
Now, as per the question, we need to find out the value of $\dfrac{{{d^2}}}{{{a^2}}}$ in terms of the square of the trigonometric function value.
We have already found out $4{d^2} = 3{a^2}$
Therefore, we get
$\dfrac{{{d^2}}}{{{a^2}}} = \dfrac{3}{4}$
The above can be further written as
$\dfrac{{{d^2}}}{{{a^2}}} = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2}$
We know that out of the given options, only $\cos {30^o} = \dfrac{{\sqrt 3 }}{2}$
Thus, we can write
$\dfrac{{{d^2}}}{{{a^2}}} = {\left( {\cos {{30}^o}} \right)^2} = {\cos ^2}{30^o}$

Hence, the fourth option is the correct option.

Note: For any quadratic equation, the value of its discriminant gives the nature of its roots. If the discriminant is greater than zero, then the roots are real and distinct. If the discriminant is equal to zero, then the roots are real and equal. If the discriminant is less than zero, then the roots are not real (they are complex).