Question

A line a drawn through A (4, -1) parallel to the line 3 x - 4 y + 1 = 0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.

Answer 1

$\text{The required line is parallel to}3x-4y+1=0$

$\therefore \text{Slope of the line = slope of}3x-4y+1$

$=\frac{-3}{-4}$

$tan\alpha =\frac{3}{4}$

$\Rightarrow sin\alpha =\frac{3}{5}andcos\alpha \frac{4}{5}$

$\text{The equation of line passing through A}$

$(4,-1)\text{and having slope}\frac{3}{4}is$

$\frac{x-x_1}{cos\theta }=\frac{y-y_1}{sin\theta }=r$

$\frac{x-4}{cos\alpha }=\frac{y+1}{sin\alpha }=r$

$\Rightarrow \frac{x-4}{\frac{4}{5}}=\frac{y+1}{\frac{3}{5}}=\pm 5$

$\Rightarrow x=8andy=2$

$or$

$x=0andy=-4$

$\therefore (8,2)and(0,-4)\text{are coordinates of two}$

$\text{points on the line which are at a distance of 5 units from (4,~1)}$