$a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x = 0, a_{2} x^{2} + 2 h_{2} x y + b_{2} y^{2} + 2 g_{2} x = 0$ will be right angles is

g_{1} (a_{2} + b_{2}) = g_{2} (a_{1} + b_{1})

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

g_{1} (a_{2} - b_{2}) = g_{2} (a_{1} + b_{1})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

g_{1} (a_{2} + b_{2}) = g_{2} (a_{1} - b_{1})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

g_{1} (a_{2} - b_{2}) = g_{2} (a_{1} - b_{1})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $g_{1} (a_{2} + b_{2}) = g_{2} (a_{1} + b_{1})$
$solution: a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} = - 2 g_{1} x - (1)$

$a_{2} x^{2} + 2 h_{2} x y + b_{2} y^{2} = - 2 g_{2} x - (11)$

diving equation (1) by (11)

$\frac{a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2}}{a_{2} x^{2} + 2 h_{2} x y + b_{2} y^{2}} = \frac{- 2 g_{1} x}{- 2 g_{2} x}$

$\Rightarrow a_{1} g_{2} x^{2} + 2 h_{1} g_{2} x y + b_{1} g_{2} y^{2} = a_{2} g_{1} x^{2} + 2 h_{2} g_{1} x y + b_{2} g_{1} y^{2}$

$\Rightarrow x^{2} (a_{1} g_{2} - a_{2} g_{1}) + y^{2} (b_{1} g_{2} - b_{2} g_{1}) + 2 x y (h_{1} g_{2} - h_{2} g_{1}) = 0$

since lines are mutually perpendicular.

so co-efficient of $x^{2} +$ co-efficient of $y^{2} = 0$

$\Rightarrow (a_{1} g_{2} - a_{2} g_{1}) + (b_{1} g_{2} - b_{2} g_{1}) = 0$

$\Rightarrow a_{1} g_{2} + b_{1} g_{2} - a_{2} g_{1} - b_{2} g_{1} = 0$

$\Rightarrow g_{2} (a_{1} + b_{1}) = g_{1} (a_{2} + b_{2})$

Answer: option (A)

Suggest Corrections

Similar questions

z ¯ ¯ ¯ z + z_{1} + a_{1} ¯ ¯ ¯ z + b_{1} = 0, b_{1} \in R

z ¯ ¯ ¯ z + z_{2} +_{2} a_{2} + b_{2} = 0, b_{2} \in R

z ¯ ¯ ¯ z + z {¯ ¯ ¯ a}_{1} + z {¯ ¯ ¯ a}_{1} + b_{1} = 0, b_{1} \in R

z ¯ ¯ ¯ z + z {¯ ¯ ¯ a}_{2} + z {¯ ¯ ¯ a}_{2} + b_{2} = 0, b_{2} \in R

R e (a_{1} {¯ ¯ ¯ a}_{2}) = b_{1} + b_{2}

b_{1}, b_{2}, . . . . .

b_{1} + b_{2} = 1

\infty \sum k = 1 b_{k} = 2

b_{2} < 0

b_{1}

The condition for a pair of equations $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ to be parallel is

Unique point is obtained for the pair of equations $a_{1}$ x + $b_{1}$ y + $c_{1}$ = 0 and $a_{2}$ x + $b_{2}$ y + $c_{2}$ = 0 if

Join BYJU'S Learning Program