Question

If a straight line $L$ with negative slope passes through $P(1,4)$ and meets the coordinate axes at A and $B$ such that area of triangle $OAB$ is minimum ( $O$ is origin), then which of the following statement(s) is(are) correct?

• A. y-intercept of line $L$ equals 2.
• B. Distance of line $L$ from $(\sqrt{4},\sqrt{17})$ equals 1 .
• C. Area of triangle $OAB$ equals 16.
• D. Radius of the circle inscribed in triangle $OAB$ equals $5-\sqrt{17}$.

Answer 1

Answer: (A), (D)

Let the line be
$\frac{x}{a}+\frac{y}{b}=1,a,b>0$
where $\frac{1}{a}+\frac{4}{b}=1\Rightarrow \frac{4}{b}=1-\frac{1}{a}\Rightarrow \frac{4}{b}=\frac{a-1}{a}$
$b=\frac{4a}{a-1}$
Now, $2A=ab=a\left(\frac{4a}{a-1}\right)=\frac{4\left(a^2-1+1\right)}{a-1}=4\left[(a+1)+\frac{1}{a-1}\right]$

$2\frac{dA}{da}=4\left[1-\frac{1}{(a-1{)}^2}\right]=0$
$a-1=1\text{ or }-1$
$a=2;b=8,m=-4$
$A{]}_{\min }=\frac{16}{2}=8$
Equation is $(y-4)=-4(x-1)\Rightarrow y-4=-4x+4\Rightarrow 4x+y=8$
Also, $r=\frac{\Delta }{s}=\frac{\frac{1}{2}\times 2\times 8}{\frac{2+8+\sqrt{68}}{2}}=\frac{8}{5+\sqrt{17}}=5-\sqrt{17}$
Now verify alternatives.