First slide

Pair of straight lines

Question

If the pair of lines which joins the origin to the point of intersection of $a x^{2} + 2 h x y + b y^{2} + 2 g x = 0$ , $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x = 0$ are at right angles then

Moderate

A

$\frac{g}{g_{1}} = \frac{a_{1} + b_{1}}{a + b}$
B

$\frac{g}{g_{1}} = \frac{a + b}{a_{1} + b_{1}}$
C

$\frac{h}{h_{1}} = \frac{a + b}{a_{1} + b_{1}}$
D

$\frac{h}{h_{1}} = \frac{a_{1} + b_{1}}{a + b}$

Solution

$\frac{a x^{2} + 2 h x y + b y^{2}}{2 g} = \frac{a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2}}{2 g_{1}} = - x \Rightarrow g_{1} (a x^{2} + 2 h x y + b y^{2} + 2 g x)$
$- g (a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x) = 0$
coefficient of $x^{2}$ + coefficient of $y^{2}$ = 0
$\Rightarrow g_{1} (a + b) = g (a_{1} + b_{1})$

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