Question

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

• A. a circle
• B. a parabola
• C. Ellipse
• D. a hyparabola

Answer 1

Answer: (B)

a parabola

Line passing through (1,1) is
$y-1=mx-m$
$y=mx+1-m$
So, y intercept $=1-m$ & x intercept $=\frac{m-1}{m}$
as given $h=\frac{1-m+\frac{m-1}{m}}{2}=\frac{m-m^2+m-1}{2m}$
$h=\frac{-(m-1{)}^2}{2m}$ ---(1)
& $k=\left[\frac{(m-1)}{m}\right(1-m){]}^{\frac{1}{2}}=(m-1)\sqrt{\frac{-1}{m}}$
$\Rightarrow k^2=\frac{(m-1{)}^2}{-m}=\frac{-2h}{-1}=2h$ [from 1]
$\Rightarrow k^2=2h$
i.e. point (h,k) lies on a curve $y^2=2x$ parabola