Question

Which of the following equations represents a line that is perpendicular to y=−2x+4 and passes through the point (4, 2)?

a) y=x+4
b) y−3x
c) y=−3x+2
d) y=−2x

Answer 1

The equation representing a line perpendicular to
`\(y = -2x + 4\)` and passing through (4, 2) is
`\(y = (1)/(2)x + 4\)`, making the slope the negative reciprocal of -2. So the correct option is A.

To find a line perpendicular to
`\(y = -2x + 4\)` and passing through the point
`\((4, 2)\),` we need to consider the negative reciprocal of the slope of the given line.

The given line has a slope of -2. The negative reciprocal of -2 is
`\(1/2\).`

Now, we can use the point-slope form of the equation of a line:

`\[y - y_1 = m(x - x_1)\]`

where \(m\) is the slope and
`\((x_1, y_1)\)`is a point on the line.

In this case,
`\(m = 1/2\)` and
`\((x_1, y_1) = (4, 2)\).`

So, the equation becomes:

`\[y - 2 = (1)/(2)(x - 4)\]`

Now, let's simplify this equation:

`\[y - 2 = (1)/(2)x - 2\]`

Add 2 to both sides:

`\[y = (1)/(2)x\]`

Therefore, the correct option is:

a)
`\(y = (1)/(2)x + 4\)`