Consider a line with slope m and y-intercept $( 0,6 )$ Write the distance d between the point $( 5,1 )$ and the line as a function of m.
A) $\frac { 5 | m - 1 | } { \sqrt { m ^ { 2 } + 1 } }$
B) $\frac { 5 | m + 1 | } { \sqrt { m ^ { 2 } - 1 } }$
C) $\frac { 5 | m + 1 | } { \sqrt { m ^ { 2 } + 1 } }$
D) $\frac { 5 | m + 5 | } { \sqrt { m ^ { 2 } + 1 } }$
E) $\frac { | m + 1 | } { \sqrt { m ^ { 2 } + 1 } }$

Question

Consider a line with slope m and y-intercept $( 0,6 )$ Write the distance d between the point $( 5,1 )$ and the line as a function of m. 
A) $\frac { 5 | m - 1 | } { \sqrt { m ^ { 2 } + 1 } }$
B) $\frac { 5 | m + 1 | } { \sqrt { m ^ { 2 } - 1 } }$
C) $\frac { 5 | m + 1 | } { \sqrt { m ^ { 2 } + 1 } }$
D) $\frac { 5 | m + 5 | } { \sqrt { m ^ { 2 } + 1 } }$
E) $\frac { | m + 1 | } { \sqrt { m ^ { 2 } + 1 } }$

Quizplus · Accepted Answer

The standard form of the equation of a line is y = mx + b, where m is the slope and b is the y-intercept. So the equation of the given line is y = mx + 6. The distance between a point (x1, y1) and a line Ax + By + C = 0 is given by:

d = |Ax1 + By1 + C| / sqrt(A^2 + B^2)

We can rewrite the equation of the line as -mx + y - 6 = 0 and substitute in the values of x1 = 5 and y1 = 1 to get:

d = |(-m)(5) + (1) - 6| / sqrt((-m)^2 + 1^2)

Simplifying this gives us:

d = |5 - m| / sqrt(m^2 + 1)

Therefore, the best choice is (C).

Consider a Line with Slope M and Y-Intercept (0,6)( 0,6 )(0,6)

Consider a Line with Slope M and Y-Intercept $( 0,6 )$