To find the point on the line defined by the equation y = 4x + 5 that is closest to the origin (0, 0), we can use the concept of the distance between a point and a line.
The distance d from a point (x_0, y_0) to a line in the form of Ax + By + C = 0 is given by the formula:
d = rac{|Ax_0 + By_0 + C|}{ ext{sqrt}(A^2 + B^2)}
First, we need to rewrite the line equation in the standard form:
y = 4x + 5 means:
4x – y + 5 = 0
In this case:
- A = 4
- B = -1
- C = 5
Now, we can substitute the coordinates of the origin (0, 0) into the distance formula:
d = rac{|4(0) – 1(0) + 5|}{ ext{sqrt}(4^2 + (-1)^2)} = rac{5}{ ext{sqrt}(16 + 1)} = rac{5}{ ext{sqrt}(17)}
This distance indicates how far the origin is from the line, but we want the closest point on the line instead. The coordinates of the closest point can also be found by using the projection method.
To determine the closest point, we set the slope of the line perpendicular to our original line (which has a slope of 4) to be -1/4. Using point-slope form, the equation of the perpendicular line through the origin (0, 0) is:
y – 0 = - rac{1}{4}(x – 0) => y = - rac{1}{4}x
Next, we solve the system of equations formed by:
- y = 4x + 5
- y = - rac{1}{4}x
Setting the two equations equal to each other:
4x + 5 = - rac{1}{4}x
To eliminate the fraction, multiply every term by 4:
16x + 20 = -x
Now, bring all terms involving x to one side:
16x + x + 20 = 0 => 17x + 20 = 0 => 17x = -20 => x = - rac{20}{17}
Now substitute x back into the line equation to find y:
y = 4(- rac{20}{17}) + 5 = - rac{80}{17} + rac{85}{17} = rac{5}{17}
Thus, the closest point on the line y = 4x + 5 to the origin is:
(- rac{20}{17}, rac{5}{17})