First slide

Equation of line in Straight lines

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

Moderate

A

a circle
B

a parabola
C

a straight line
D

a hyperbola

Solution

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

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