Question

The centre of a circle is $(x-2,x+1)$ and it passes through the points $(4,4)$ Find the value ( or values ) of $x$, if the diameter of the circle is of length $2\sqrt{5}$ units.

• A. $1$ or $3$
• B. $-1$ or $4$
• C. $5$ or $4$
• D. $3$ or $-2$

Answer 1

The correct option is C $5$ or $4$

Radius of the circle $=$ dist. between the center and given pt. on the circle.

Distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ can be calculated using the formula

$\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
Distance between the points $(x-2,x+1)$ and D $(4,4)$

$=\sqrt{{(4-x+2)}^2+{(4-x-1)}^2}$

$=\sqrt{{(6-x)}^2+{(3-x)}^2}$

$=\sqrt{36+x^2-12x+9+x^2-6x}=\sqrt{2x^2-18x+45}$

Given, diameter $=2\sqrt{5}$ $\Rightarrow$ Radius $=\sqrt{5}$

$\Rightarrow \sqrt{2x^2-18x+45}=\sqrt{5}$
Squaring both sides,
$2x^2-18x+45=5$
$2x^2-18x+40=0$
$x^2-9x+20=0$
$(x-5)(x-4)=0$
$x=4$ or $5$