Find the condition that the lines joining the origin to the points of intersection of the curves $a x^{2} + 2 h x y + b y^{2} 2 g x = 0$ and $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x = 0$ be perpendicular.

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Solution

We know that the combined equation of the lines through the origin is homogeneous. Hence eliminating the first degree terms of x between the given equations we shall obtain homogeneous equation which will represent the required lines. Multiplying the first equation by $g_{1}$ and $2$ nd by g and subtracting, we get $g_{1} (a x^{2} + 2 h x y + b y^{2} + 2 g x) - g (a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x) = 0$
or $(g_{1} a - a_{1} g) x^{2} + 2 x y (h g_{1} - h_{1} g) + (g_{1} b - b_{1} g) y^{2} = 0$
If the equation represents two perpendicular lines, then
coeff. of $x^{2} +$ coeff. of $y^{2} = 0$
$∴ g_{1} a - a_{1} g + g_{1} b - b_{1} g = 0$
$∴ (a + b) g_{1} = (a_{1} + b_{1}) g$ is the required condition.

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Q. Pair of lines joining origin to the points of intersection of two curves

a x^{2} + 2 h x y + b y^{2} + 2 g x = 0

and

a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x = 0

will be at right angled to each other if

\frac{a_{1} + b_{1}}{a + b} =

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

The line joining the origin to the points of intersection of the curves $a x^{2} + 2 h x y + b y^{2} + 2 g x = 0$ and $a^{^{'}} x^{2} + 2 h^{^{'}} x y + b^{^{'}} y^{2} + 2 g^{^{'}} x = 0$ will be mutually perpendicular, if

Q. The condition that one of the lines represented by

a x^{2} + 2 h x y + b y^{2} = 0

is perpendicular to one of the lines represented by

a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} = 0

Q. Show that the straight lines joining the origin to the other two points of intersection of the curves whose equations are

a x^{2} + 2 h x y + b y^{2} + 2 g x = 0

and

a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x = 0

will be at right angles if

g (a^{'} + b^{'}) - g^{'} (a + b) = 0 .

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Find the condition that the lines joining the origin to the points of intersection of the curves ax2+2hxy+by22gx=0 and a1x2+2h1xy+b1y2+2g1x=0 be perpendicular.

Find the condition that the lines joining the origin to the points of intersection of the curves $a x^{2} + 2 h x y + b y^{2} 2 g x = 0$ and $a_{1} x^{2} + 2 h_{1} x y + b_{1} y^{2} + 2 g_{1} x = 0$ be perpendicular.