Question

The line segment joining the points (4, 7) and $(-2,-1)$ is a diameter of a circle. If the circle intersects the x-axis at A and B, then AB is equal to

• A. 4
• B. 5
• C. 6
• D. 7
• E. 8

Answer 1

Answer: (E)

$\because$ (4, 7) and $(-2,-1)$ are end points of circle.

$\Rightarrow$ Centre of circle
$=\left(\frac{4-2}{2},\frac{7-1}{2}\right)=(1,3)$
$\therefore$ Slope of $AC=\frac{-1-0}{-2-a}=-\frac{1}{2+a}$ and slope of $AD=\frac{7-0}{4-a}=\frac{7}{4-a}$
Since, $\Delta CAD$ is right angle as CD is diameter.
$\Rightarrow$ (slope of AC (slope of AD)
$=-1$
$\Rightarrow$ $\left(\frac{1}{2+a}\right)\left(\frac{7}{4-a}\right)=-1$
$\Rightarrow$ $\frac{7}{-a^2+2a+8}=-1$
$\Rightarrow$ $a^2-2a-15=0$
$\Rightarrow$ $a=-3,5$ Similarly, $b=-3,5$
$\therefore$ Point $A=(-3,0)$ and point $B=(5,0)$
$\therefore$ $AB=\sqrt{{(-3-5)}^2+{(0-0)}^2}$
$=\sqrt{{(-8)}^2}=8$