The number of lines that are parallel to 2x + | vedclass

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(B) The equation of any line parallel to $2x + 6y + 7 = 0$ is of the form $2x + 6y + k = 0$.This line intersects the $x$-axis at $A\left(-\frac{k}{2},

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$2$

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(B) The equation of any line parallel to $2x + 6y + 7 = 0$ is of the form $2x + 6y + k = 0$.This line intersects the $x$-axis at $A\left(-\frac{k}{2}, 0\right)$ and the $y$-axis at $B\left(0, -\frac{k}{6}\right)$.The length of the intercept between the axes is given as $AB = 10$.Using the distance formula,$AB = \sqrt{\left(-\frac{k}{2} - 0\right)^2 + \left(0 - \left(-\frac{k}{6}\right)\right)^2} = 10$.$\sqrt{\frac{k^2}{4} + \frac{k^2}{36}} = 10$.$\sqrt{\frac{9k^2 + k^2}{36}} = 10 \Rightarrow \sqrt{\frac{10k^2}{36}} = 10$.$\frac{\sqrt{10}|k|}{6} = 10 \Rightarrow |k| = \frac{60}{\sqrt{10}} = 6\sqrt{10}$.Thus,$k = \pm 6\sqrt{10}$.Therefore,there are $2$ such lines: $2x + 6y + 6\sqrt{10} = 0$ and $2x + 6y - 6\sqrt{10} = 0$.

The number of lines that are parallel to $2x + 6y + 7 = 0$ and have an intercept of length $10$ between the coordinate axes is

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