Question

The equation y = 2x + 7 represents a northern path on a map laid on a coordinate grid plane. a) Find an equation for a path that passes through the point (6,3) and is parallel to this northern path. b) Find an equation for a path that passes through the point (6,3) and is perpendicular to this northern path.

Answer 1

SOLUTION

GIVEN

The equation y = 2x + 7 represents a northern path on a map laid on a coordinate grid plane.

TO DETERMINE

a) The equation for a path that passes through the point (6,3) and is parallel to this northern path.

b) The equation for a path that passes through the point (6,3) and is perpendicular to this northern path.

EVALUATION

Here the given equation of the northern path on a map laid on a coordinate grid plane is

y = 2x + 7

Which can be rewritten as

2x - y + 7 = 0 - - - - (1)

(a) We have to find the equation of the path parallel to this northern path.

Let the equation of the line parallel to this northern path is

2x - y + c = 0 - - - - (2)

Now Equation 2 passes through the point (6,3)

So we have

$\displaystyle \sf{ (2 \times 6) - 3 + c = 0}$

$\displaystyle \sf{ \implies 12 - 3 + c = 0}$

$\displaystyle \sf{ \implies 9 + c = 0}$

$\displaystyle \sf{ \implies c = - 9}$

Hence the required equation of the line parallel to the northern path is

2x - y - 9 = 0

(b) We have to find the equation of the line perpendicular to the northern path

Let the equation of the line perpendicular to the northern path is

x + 2y + k = 0 - - - - - (3)

Equation 3 passes through the point (6,3)

So we have

$\displaystyle \sf{ \implies 6 + (2 \times 3) + k = 0}$

$\displaystyle \sf{ \implies 6 +6+ k = 0}$

$\displaystyle \sf{ \implies 12+ k = 0}$

$\displaystyle \sf{ \implies k = - 12}$

Hence the required equation of the line perpendicular to the northern path is

x + 2y - 12 = 0

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