How do you find the perpendicular distance between the straight lines #y=2x+7# and #y=2x+2#?

Answer 1

#5/ sqrt 5 = sqrt 5#

#r : y = 2x + 7#
#s : y = 2x + 2#
#r# and #s# are parallel, and both are perpendicular to
#t : y = -1/2 x + b#
Let #b = 0# and find #r cap t = P, s cap t = Q#
#2x + n = -1/2 x#
#4x + 2n = -x#
#x = - (2n) / 5#
#y = -1/2 x = n/5#
#|PQ|^2 = (-(2n_1)/5 + (2n_2)/5)^2 + ((n_1)/5 - (n_2)/5)^2#
#|PQ|^2 = 4/25 (n_2 - n_1)^2 + 1/25 (n_1 - n_2)^2#
#|PQ| = sqrt {1/5} |n_1 - n_2|#
#n_1 = 7, n_2 = 2#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the perpendicular distance between two parallel lines, ( y = mx + c_1 ) and ( y = mx + c_2 ), where ( m ) is the slope and ( c_1 ) and ( c_2 ) are the y-intercepts, we can use the following formula:

[ \text{Distance} = \frac{|c_2 - c_1|}{\sqrt{1 + m^2}} ]

For the given lines ( y = 2x + 7 ) and ( y = 2x + 2 ), both lines have the same slope ( m = 2 ).

The y-intercepts are ( c_1 = 7 ) for the first line and ( c_2 = 2 ) for the second line.

Substituting these values into the formula:

[ \text{Distance} = \frac{|2 - 7|}{\sqrt{1 + 2^2}} ] [ \text{Distance} = \frac{5}{\sqrt{5}} ] [ \text{Distance} = \frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} ] [ \text{Distance} = \frac{5\sqrt{5}}{5} ] [ \text{Distance} = \sqrt{5} ]

So, the perpendicular distance between the lines ( y = 2x + 7 ) and ( y = 2x + 2 ) is ( \sqrt{5} ) units.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7