If sin alpha and cos alpha are the roots of t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given that $\sin \alpha$ and $\cos \alpha$ are the roots of the quadratic equation $ax^2 + bx + c = 0$.From the properties of roots,the sum of roo

Vedclass · Accepted Answer

$a^2 - b^2 + 2ac = 0$

Vedclass · Answer

(A) Given that $\sin \alpha$ and $\cos \alpha$ are the roots of the quadratic equation $ax^2 + bx + c = 0$.From the properties of roots,the sum of roots is $\sin \alpha + \cos \alpha = -\frac{b}{a}$ and the product of roots is $\sin \alpha \cos \alpha = \frac{c}{a}$.We know the identity $\sin^2 \alpha + \cos^2 \alpha = 1$.This can be written as $(\sin \alpha + \cos \alpha)^2 - 2\sin \alpha \cos \alpha = 1$.Substituting the values of the sum and product of roots:$(-\frac{b}{a})^2 - 2(\frac{c}{a}) = 1$$\frac{b^2}{a^2} - \frac{2c}{a} = 1$Multiplying both sides by $a^2$:$b^2 - 2ac = a^2$Rearranging the terms,we get $a^2 - b^2 + 2ac = 0$.

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $ax^2 + bx + c = 0$,then:

Similar Questions

The coefficient of $x$ in the equation $x^2 + px + q = 0$ was taken as $17$ in place of $13$. Its roots were found to be $-2$ and $-15$. The roots of the original equation are:

The roots of the equation $a(x^2 + 1) - (a^2 + 1)x = 0$ are

$P$ can complete a piece of work in $16$ days,which $Q$ can complete in $32$ days. If $P$ and $Q$ work on alternate days,find the relationship between Quantity $I$ (Total days taken if $P$ starts) and Quantity $II$ (Total days taken if $Q$ starts).

Solve the given two equations and select the correct option.
$I.$ $2x^2 + 23x + 63 = 0$
$II.$ $4y^2 + 19y + 21 = 0$

The condition for the roots of the equation $(c^2 - ab)x^2 - 2(a^2 - bc)x + (b^2 - ac) = 0$ to be equal is

Vedclass Test Series

Exam Paper Generator

Online Exam Module