Question

Through the point A(2, 3) a straight line is drawn making + ve intercepts on the coordinates axes. If the area of the triangle so formed is least then the ratio of the x, y intercepts is

• A. 2 : 3
• B. 3 : 2
• C. 4 : 9
• D. 9 : 4

Answer 1

Answer: (A)

2 : 3

Equation of straight line passing through (2,3) and having slope m is given
$y-3=m(x-2)$
Since, this line intersects coordinate axes at A and B.
So, coordinates of A are $(\frac{2m-3}{m},0)$
Coordinates of B are $(0,-2m+3)$
Area of $△AOB=-\frac{1}{2}\frac{(2m-3{)}^2}{m}$
$A=f\left(m\right)=-\frac{1}{2}\left(\frac{9m^2-24m+16}{m}\right)$
For maxima or minima,
$f^{\prime }\left(m\right)=0$
$\Rightarrow \frac{4m^2-9}{m^2}=0$
$\Rightarrow m=\pm \frac{3}{2}$
$f^{\prime \prime }\left(m\right)=-\frac{18}{m^3}$
$f^{\prime \prime }\left(m\right)>0$ at $m=-\frac{3}{2}$
So, area of triangle is minimum at $m=-\frac{3}{2}$
So, x-intercept $=4$ and y-intercept $=6$
So, $x:y=2:3$