Question

The radius of the smallest circle which touches the straight-line $3x-y=6$ at $(1,-3)$ and also touches the line $y=x$ is
(Compute upto one place of decimal only)

• A. $2$ units
• B. $1.5$ units
• C. $1.8$ units
• D. $1.2$ units

Answer 1

The correct option is B $1.5$ units

$\because 3x-y=6$ and $y=x$ touches the same circles and are not parallel to each other
So, we can say that tangents are drawn from a point $P$ to the circles
$\therefore P\equiv (3,3)$
Now length of the tangent is
$L=\sqrt{(3-1{)}^2+(3+3{)}^2}=2\sqrt{10}$ units
Now if the angle between the line $2\theta$ half angle between the lines
$\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\Rightarrow \frac{3-1}{1+3}=\frac{2\tan \theta }{1-{\tan }^2\theta }\Rightarrow \tan \theta =\pm \sqrt{5}-2$
for acute angle $\tan \theta =\sqrt{5}-2$
Now $\tan \theta =\frac{r}{L}$
$\Rightarrow \sqrt{5}-2=\frac{r}{2\sqrt{10}}\Rightarrow r=10\sqrt{2}-4\sqrt{10}\simeq 1.5\text{units}$