Line ℓ passes through (-2, 5) and (4, -7). Write an equation for the line that passes through (2, 9) and is perpendicular to line ℓ. Leave in slope-intercept form.

Answer 1

l: m=(y₂-y₁)/(x₂-x₁) - (-7-5)/(4-(-2)) = -12/6 = -2

For the perpendicular line:

m_p=-1/m = -1/-2 = 1/2

y_p = m_p(x-2)+9 = (x-2)/2+9 = x/2+8

y_p = x/2+8

Answer 2

To find the equation of a line that passes through (2,9)(2, 9) and is perpendicular to the line ℓ\ell passing through (−2,5)(-2, 5) and (4,−7)(4, -7), follow these steps:

Step 1: Find the slope of line ℓ\ell

We first need to find the slope of line ℓ\ell. The formula for the slope of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}Substitute the points (−2,5)(-2, 5) and (4,−7)(4, -7) into the formula:

m=−7−54−(−2)=−126=−2m = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2So, the slope of line ℓ\ell is m=−2m = -2.

Step 2: Find the slope of the perpendicular line

Lines that are perpendicular have slopes that are negative reciprocals of each other. The negative reciprocal of −2-2 is 12\frac{1}{2}.

So, the slope of the line that is perpendicular to ℓ\ell is m=12m = \frac{1}{2}.

Step 3: Use point-slope form to write the equation

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

y−y1=m(x−x1)y - y_1 = m(x - x_1)where mm is the slope and (x1,y1)(x_1, y_1) is a point on the line.

We are given the point (2,9)(2, 9) and the slope m=12m = \frac{1}{2}. Substituting these values into the point-slope form:

y−9=12(x−2)y - 9 = \frac{1}{2}(x - 2)Step 4: Simplify to slope-intercept form

Distribute 12\frac{1}{2} on the right-hand side:

y−9=12x−1y - 9 = \frac{1}{2}x - 1Next, add 9 to both sides to solve for yy:

y=12x+8y = \frac{1}{2}x + 8Final Answer:

The equation of the line passing through (2,9)(2, 9) and perpendicular to line ℓ\ell is:

y=12x+8y = \frac{1}{2}x + 8

Answer 3

Slope of l: m₁ = (- 7 - 5)/(4 + 2) = - 2; m₂ = - 1/m₁ = 1/2.

Equation of 2nd line: y = 9 + 1/2(x - 2); y = 1/2x + 8