Question

A straight line slides along the axes (oblique) of x and y, and the difference of the intercepts is
always proportional to the area it encloses. Show that the line always passes through a fixed point

Answer 1

Answer on Question #61284 - Math - Analytic Geometry

Question

A straight line slides along axes (oblique) (of x and y, and the difference of the intercepts is always proportional to the area it encloses. Show that the line always passes through a fixed point.

Solution

$$\frac {x}{a} + \frac {y}{b} = 1$$

Difference of the intercepts is $a - b$ , area is $\frac{ab}{2}$ .

It is given that

$$\frac {a - b}{a b} = \text {constant} = \frac {1}{k},$$

where $k$ is fixed.

Multiplying both sides of (2) by $k$

$$k \frac {a - b}{a b} = 1$$

Then

$$\frac {k \cdot a}{a \cdot b} + \frac {- k \cdot b}{a \cdot b} = 1$$

$\frac{-k}{a} + \frac{k}{b} = 1 \Rightarrow (-k, k)$ is a point of the straight line according to the equation (1).

Thus, the line always passes through the fixed point $(-k,k)$ .

www.AssignmentExpert.com