Question

Find the value of $x_i$, if the distance between the points $(x_i,2)$ and $(3,4)$ is $8$.

• A. $3\pm 2\sqrt{15}$
• B. $3\pm \sqrt{15}$
• C. $2\pm 3\sqrt{15}$
• D. $2\pm \sqrt{15}$

Answer 1

The correct option is A $(3\pm 2\sqrt{15})$

Given points $(x_1,y_1)=(x_i,2)$ & $B(x_2,y_2)=(3,4)$
Let $d$ is the distance between point $A$ & $B$ . So $d=8$
$\Rightarrow d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$
$8=\sqrt{(3-x_i{)}^2+(4-2{)}^2}$
taking square on both side,
$64=(3-x_i{)}^2+(4-2{)}^2$
$64=9-6x_i+(x_i{)}^2+(2{)}^2$
$(x_i{)}^2-6x_i-51=0$
Now, discriminant $D=\sqrt{b^2-4ac}$
Here $b=-6$ , $a=1$ ,$c=-51$
so, $d=\sqrt{(-6{)}^2-4(1\left)\right(-51)}=\sqrt{36+204}$
$d=\sqrt{240}>0$
Thus, $x_i=\frac{-b\pm \sqrt{d}}{2a}$
$=\frac{-(-6)\pm \sqrt{240}}{2\left(1\right)}$
$=\frac{6\pm 4\sqrt{15}}{2}$
$\therefore x_i=3\pm 2\sqrt{15}$