Give your answers in the form ax + by + c = 0, where a,b,c are integers.

Question

The Points P and Q have coordinates P(4,0) and Q(-5,7).Find the equation of the line with is perpendicular to PQ and passes through P.

Answer 1

One Line [A} is perpendicular to a Second Line [B] if it has a slope that is the "NEGATIVE RECIPROCAL" of the first one.

A negative reciprocal is the fraction you multiply the first fraction to get -1.

Basically, flip the fraction and change its sign (the negative reciprocal of 2/3 is -3/2)

So its a 2 step process:

Find the original slope.

The find the y-intercept (the place on the y axis where x is 0).

So,

slope is rise (difference in Y-coordinates) over run (difference in X-coordinates)

7 - 0 = 7 That's the original slope

-5 - 4 -9

The slope of the perpendicular line (flip and change sign) is 9

7

now you need a line that has that new slope where one answer is

(4,0)

to do this you can use the "POINT-SLOPE" formula

(Y - Y_p) = m(X - X_p)

Y and X are just variables

Y_p and X_p are the x and y for point P(4, 0)

m is the SLOPE

now just plug and play

(Y - 0) = 9 (X-4)

7

Y= 9X - 4

7 7

subtract the Y to move it on the same side as the rest of the equation...

9X - Y - 4 = 0

7 7

Now make integers out of everything, multiply everything by 7.

If you multiply everything in an equation by the same number, the equation is still true.

9x - 7y - 4 = 0



Answer 2

Hello, Cordell,

Let6's calculate the slope of the line formed by the two given points. Slope is m in the general equation y=mx+b. It can be calculated as the "Rise" over the "Run." The rise is the change in y over the interval of x, which is the run.

The line between these two points would "rise" by (0 - 7) = -7. The x interval is (4 - (- 5)) = 9. The slope is therefore -(7/9). The slope of a perpendicular line is the negative inverse of this number, or (9/7). So the perpendicular line would be of the form y = (9/7)x + c

We are told that the new line also passes through P (4,0). Use this point to calculate c:

0 = (9/7)*(4) + c

0 = (36/7) + c

c = - (36/7)

The equation is thus y = (9/7)*x - (36/7)

The format is requested to be ax + by + c = 0

(9/7)x -1y - (36/7) = 0

Bob

[Update]: David W. noted that my result did not establish am b, and c as integers. Multiply the equation by 7 to obtain his correct answer in the comments section.

9x - 7x - 36 = 0