Question

$A$ and $B$ are two points on the circle $x^2+y^2=1$. lf the $x$ co-ordinates of $A$ and $B$ are the roots of the equation $x^2+ax+b=0$ and the $y$
coordinates $ofA$ and $B$ are the roots of the equation $y^2+by+a=0$ then the equation of the line $AB$ is

• A. $ax+by=0$
• B. $ax+by+1=0$
• C. $bx+ay+a+b=0$
• D. $ax+by+a+b+1=0$

Answer 1

Answer: (D)

$ax+by+a+b+1=0$

Let, $x^2+ax+b=0hasroots{x}_1,x_2$
$\therefore x_1+x_2=-a$
$x_1{x}_2=b$
and $y^2+by+a=0hasroots{y}_1,x_2)$
and $A(x_1,y_1)andB(x_2,y_2)$ are point on the circle $x^2+y^2=1$
$\therefore x_1^2+y_1^2=1=x_2^2+y_2^2\Rightarrow x_1^2-x_2^2=y_2^2-y_1^2$
$\therefore e{q}^{n}ofABis$
$y-y_1=\frac{y_1-y_2}{x_1-x_2}(x-x_1)$
$y(x_1-x_2)-y_1{x}_1+x_2{y}_1=\left(y_1{y}_2\right)k-x_1{y}_1+x_1{y}_2$
$y(x_1-x_2)-x(y_1-y_2)=x_1{y}_2-x_2{y}_1$
$y+x\left(\frac{x_1+x_2}{y_1+y_2}\right)=\frac{x_1{y}_2-x_2{y}_1}{x_1-x_2}$
$\Rightarrow by+ax=b\frac{(x_1{y}_2-x_2{y}_1)}{x_1{x}_2}$
$\Rightarrow by+ax=b\frac{b(x_1\sqrt{1-x_2^2}-x_2(\sqrt{1-x_1^2})}{x_1{x}_2}$
$\Rightarrow by+ax=b\frac{(\sqrt{x_1^2-b^2}-\sqrt{x_2^2-b^2})}{x_1{x}_2}$
$\Rightarrow ax+by=-a-b-1$