Answer to Question #252728 in Analytic Geometry for ling

Question

3. Find the equation of the line parallel to 5x + 6y – 10 = 0 at a distance 61 from (-3,7).

4. Find the equation of the line parallel to 3x + 4y = 20 and at a distance of 5 units from this

line.

Answer 1

3. The equation of the line parallel to $5x + 6y – 10 = 0$ is

$5x + 6y +b = 0$

Distance from point $(-3,7)$ to the line is

$d=\dfrac{|5(-3)+6(7)+b|}{\sqrt{5^2+6^2}}=61$

$|b-27|=61\sqrt{61}$

$b=27+61\sqrt{61}$

The equation of the line is

$5x + 6y +27+61\sqrt{61} = 0$

$b=27-61\sqrt{61}$

The equation of the line is

$5x + 6y +27-61\sqrt{61} = 0$

4. The equation of the line parallel to $3x + 4y = 20$ is

$3x + 4y = b$

Take the point $(4, 2)$

$3(4) + 4(2) = 20, True$

Distance from point $(4, 2)$ to the line $3x + 4y = b$ is

$d=\dfrac{|3(4)+4(2)-b|}{\sqrt{3^2+4^2}}=5$

$|20-b|=25$

$b=-5$

The equation of the line parallel to $3x + 4y = 20$ is

$3x + 4y = -5$

$b=45$

The equation of the line parallel to $3x + 4y = 20$ is

$3x + 4y = 45$