Answer:
Explanation:
To find the distance between a line and a point, we need to find the perpendicular distance from the point to the line. Here's the step-by-step process:
Find the equation of the line perpendicular to ℓ that passes through point F(-1, 7): Since the slope of line ℓ is -1, the slope of the perpendicular line will be the negative reciprocal of -1, which is 1. To find the equation of the line, we need a point and the slope. Point F(-1, 7) will be used, and the slope is 1.
Using the point-slope form of a line, we can write the equation of the perpendicular line passing through point F as: y - 7 = 1(x + 1). Simplifying, we get: y = x + 8.
To find the point of intersection between the two lines, we can substitute the equation of line ℓ into the equation of the perpendicular line:
-x + 4 = x + 8
Solving for x, we get x = 6. Substituting x = 6 into the equation of line ℓ, we get:
y = -6 + 4 = -2
So, the point of intersection between the two lines is (6, -2).
To find the distance between the point F and the line ℓ, we can use the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) is the point F(-1, 7) and (x2, y2) is the point of intersection (6, -2).
So, substituting the values into the formula:
distance = √((6 - (-1))^2 + ((-2) - 7)^2) = √((7)^2 + (9)^2) = √(49 + 81) = √(130) = approximately 11.3
Therefore, the distance between the line ℓ and point F is approximately 11.3 units.