Question

$AB$ is a diameter of a circle; $CD$ is a chord parallel to $AB$ and $2CD=AB$. If the tangent at $B$ meets the line $AC$ produced at $E$, then $AE$ is equal to

• A. $2AB$
• B. $3AB$
• C. $AB+CD$
• D. $\sqrt{2}AB$

Answer 1

The correct option is A $2AB$

Let the equation of the circle be $x^2+y^2=a^2$ and $AB$ be the diameter along x-axis
with $A(a,0)$ and $B(-a,0).(Fig.16.38)$
If $OL$ is perpendicular from the centre $0$ of the circle on $CD$ then as $CD$ is parallel to
$AB$ and half of $AB.$
$CL=\left(1/2\right)OA=a/2=DL$
$OL=\sqrt{(OC{)}^2-(CL{)}^2}$
$=\sqrt{a^2\frac{a^2}{4}}=\frac{\sqrt{3}a}{2}$
So the coordinates of $C$ are $(a/2.-\sqrt{3}a/2)$
Equation of $AC$ is therefore $y-0=\sqrt{3}(x-a)\left(1\right)$
Equation of the tangent at $B$ is $x=-a\left(2\right)$
So the coordinates of $E,$ the point of intersection of $\left(1\right)$
and $\left(2\right)$ are $(-a.-2\sqrt{3}a).$
Thus $(AE{)}^2=(a+a{)}^2+(2\sqrt{3}a{)}^2=16a^2$
$\Rightarrow AE=2AB$.