The straight lines represented by `(y-m x)^2=a^2(1+m^2)` and `(y-n x)^2=a^2(1+n^2)` from a rectangle (b) rhombus trapezium (d) none of these

Options

A. rectangle
B. rhombus
C. trepezium
D. None of these

Answer

2

Explanation

The straight lines represented by `(y-mx)^(2)=a^(2)(1+m^(2))` are
`y-mx=+-asqrt(1+m^(2))`
i.e., `y-mx=asqrt(1+m^(2))` (1)
and `y-mx=-asqrt(1+m^(2))` (2)
Similarly , the straight lines representd by `(y-nx)^(2)=a^(2)(1+n^(2))` are
`y-nx=asqrt(1+n^(2))` (3)
and ` y-nx=-asqrt(1+n^(2))` (4)
Since the lines (1) and (2) are parallel , the distance between thems is
`|(asqrt(1+m^(2))+asqrt(1+m^(2)))/(sqrt(1+m^(2)))| =|2a|`
Similarly , the lines (3) and (4) are parallel lines and the distance between them is `|2a|` . Since the distances between parallel lines are the same , the four lines form a rhombus.