The pair of straight lines joining the origin | vedclass

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(C) The equation of the line is $y - 2\sqrt{2}x = c$,which can be written as $\frac{y - 2\sqrt{2}x}{c} = 1$.Homogenizing the circle $x^2 + y^2 = 2$ us

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$c^2 - 9 = 0$

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(C) The equation of the line is $y - 2\sqrt{2}x = c$,which can be written as $\frac{y - 2\sqrt{2}x}{c} = 1$.Homogenizing the circle $x^2 + y^2 = 2$ using the line equation,we get:$x^2 + y^2 = 2 \left( \frac{y - 2\sqrt{2}x}{c} \right)^2$$c^2(x^2 + y^2) = 2(y^2 + 8x^2 - 4\sqrt{2}xy)$$c^2x^2 + c^2y^2 = 2y^2 + 16x^2 - 8\sqrt{2}xy$$(c^2 - 16)x^2 + 8\sqrt{2}xy + (c^2 - 2)y^2 = 0$Since the lines are at right angles,the sum of the coefficients of $x^2$ and $y^2$ must be zero:$(c^2 - 16) + (c^2 - 2) = 0$$2c^2 - 18 = 0$$c^2 = 9$$c^2 - 9 = 0$.

The pair of straight lines joining the origin to the points of intersection of the line $y = 2\sqrt{2}x + c$ and the circle $x^2 + y^2 = 2$ are at right angles,if

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