If the line joining origin to the points of intersection of the line f x-g y- and the curve x2-h x y-y2-g x-f y-0 be mutually perpendicular- thenA- -hB- -gC- - may have any valueD- -f g

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Question

If the line joining origin to the points of intersection of the line fx - gy = $λ$ and the curve $x^{2} + h x y - y^{2} + g x + f y = 0$ be mutually perpendicular, then

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Solution

The correct option is D

Making the equation of curve homogeneous with the help of equation of line $\frac{f x - g y}{λ} = 1$ and to be perpendicular to both the lines represented by this homogeneous equation.

a + b = 0 $\Rightarrow λ + g f - λ - g f = 0 \Rightarrow 0 = 0$

Hence, $λ$ may have any value.

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If the line joining origin to the points of intersection of the line fx - gy = λ and the curve x2+hxy−y2+gx+fy=0 be mutually perpendicular, then

The correct option is D Making the equation of curve homogeneous with the help of equation of line fx−gyλ=1 and to be perpendicular to both the lines represented by this homogeneous equation. a + b = 0 ⇒λ+gf−λ−gf=0⇒0=0 Hence, λ may have any value.

If the line joining origin to the points of intersection of the line fx - gy = $λ$ and the curve $x^{2} + h x y - y^{2} + g x + f y = 0$ be mutually perpendicular, then