Question

1 calculate the distance from the point a(-3,2) to the line y = 2x + 1.
my calculations
2 calculate the distance from the point a(-4,-1) to the line y = x.
my calculations
3 calculate the distance from point a(0,-3) to the line 4x - 3y - 5 = 0.
my calculations
4 calculate the distance from point a(-5,2) to the line x + 2y - 4 = 0.
my calculations
5 calculate the distance between the parallel lines y = 2x - 5 and y = 2x + 1.
my calculations
6 calculate the distance between the parallel lines 2x + 3y - 12 = 0 and 4x + 6y + 12 = 0.
my calculations

Explanation:

Problem 1: Distance from \( A(-3,2) \) to \( y = 2x + 1 \)

The formula for the distance from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is \( d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \). First, rewrite \( y = 2x + 1 \) in standard form: \( 2x - y + 1 = 0 \). Here, \( A = 2 \), \( B = -1 \), \( C = 1 \), \( x_0 = -3 \), \( y_0 = 2 \).

Step 1: Substitute into the formula

\( d = \frac{|2(-3) + (-1)(2) + 1|}{\sqrt{2^2 + (-1)^2}} \)

Step 2: Simplify numerator and denominator

Numerator: \( |-6 - 2 + 1| = |-7| = 7 \)
Denominator: \( \sqrt{4 + 1} = \sqrt{5} \)
So, \( d = \frac{7}{\sqrt{5}} = \frac{7\sqrt{5}}{5} \)

Problem 2: Distance from \( A(-4,-1) \) to \( y = x \)

Rewrite \( y = x \) as \( x - y = 0 \), so \( A = 1 \), \( B = -1 \), \( C = 0 \), \( x_0 = -4 \), \( y_0 = -1 \).

Step 1: Apply the distance formula

\( d = \frac{|1(-4) + (-1)(-1) + 0|}{\sqrt{1^2 + (-1)^2}} \)

Step 2: Simplify

Numerator: \( |-4 + 1| = |-3| = 3 \)
Denominator: \( \sqrt{2} \)
Thus, \( d = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \)

Problem 3: Distance from \( A(0,-3) \) to \( 4x - 3y - 5 = 0 \)

Here, \( A = 4 \), \( B = -3 \), \( C = -5 \), \( x_0 = 0 \), \( y_0 = -3 \).

Step 1: Substitute into the formula

\( d = \frac{|4(0) + (-3)(-3) - 5|}{\sqrt{4^2 + (-3)^2}} \)

Step 2: Simplify

Numerator: \( |0 + 9 - 5| = |4| = 4 \)
Denominator: \( \sqrt{16 + 9} = 5 \)
So, \( d = \frac{4}{5} \)

Problem 4: Distance from \( A(-5,2) \) to \( x + 2y - 4 = 0 \)

Answer:

s:

1. \( \boldsymbol{\frac{7\sqrt{5}}{5}} \)
2. \( \boldsymbol{\frac{3\sqrt{2}}{2}} \)
3. \( \boldsymbol{\frac{4}{5}} \)
4. \( \boldsymbol{\sqrt{5}} \)
5. \( \boldsymbol{\frac{6\sqrt{5}}{5}} \)
6. \( \boldsymbol{\frac{18\sqrt{13}}{13}} \)