Question

If $h$ denotes the arithmetic mean and $k$ denotes the geometric mean of the intercepts made on the coordinate axes by the lines passing through the point $(1,1)$, then the point $(h,k)$ lies on

• A. a circle
• B. a parabola
• C. a straight line
• D. a hyperbola

Answer 1

The correct option is B a parabola

Let the equation of the line be $\frac{x}{a}+\frac{y}{b}=1$
If it passes through $(1,1)$ then
$\frac{1}{a}+\frac{1}{b}=1\Rightarrow a+b=ab$ ...$\left(1\right)$
Since the line makes intercepts of length $a$ and $b$ on the coordinate axes, we have
$h=\frac{a+b}{2}$ and $k=\sqrt{ab}$
So that form $\left(1\right)$ we get $2h=k^2$
and thus the point $(h,k)$ lies on
$2x=y^2$ which is parabola