First slide

Pair of straight lines

Question

The equations $a^{2} x^{2} + 2 h (a + b) xy + b^{2} y^{2} = 0$ and ${ax}^{2} + 2 hxy + {by}^{2} = 0$

Moderate

A

two pairs of perpendicular straight lines
B

two pairs of parallel straight lines
C

two pairs of straight lines which are equally inclined to each other
D

None of these

Solution

The given equations are
$a^{2} x^{2} + 2 h (a + b) xy + b^{2} y^{2} = 0$ __(1)
and ${ax}^{2} + 2 hxy + {by}^{2} = 0$ _(2)
The equation of the bisectors of the angles between the lines represented by (1) is
$\frac{x^{2} - y^{2}}{a^{2} - b^{2}} = \frac{xy}{h (a + b)}$
or $\frac{x^{2} - y^{2}}{a - b} = \frac{xy}{h}$
which is the same as the equation of bisectors of the angles between the line pair (2). Thus, two line pairs are equally inclined to each other.

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