Question

11. A line through (3, 5) and (k, 12) is perpendicular to a line through (0, 7) and (2, 10). Find the value of k that makes the above statement true.​

Answer 1

k = 23/7

Explanation:

To solve the problem, we can use the fact that the product of the slopes of two perpendicular lines is -1. We can start by finding the slope of the line passing through the points (3, 5) and (k, 12).

slope = (12 - 5) / (k - 3)

We can simplify this expression by multiplying both numerator and denominator by -1:

slope = (-7) / (3 - k)

Now, let's find the slope of the line passing through the points (0, 7) and (2, 10):

slope = (10 - 7) / (2 - 0) = 3 / 2

Since the two lines are perpendicular, we can set the product of their slopes equal to -1:

(-7) / (3 - k) * (3 / 2) = -1

Simplifying this equation, we get:

(7/2) * (3 - k) = 1

Multiplying both sides by 2/7, we get:

3 - k = 2/7

Subtracting 3 from both sides, we get:

-k = 2/7 - 3 = -21/7 - 2/7 = -23/7

Finally, dividing by -1, we get:

k = 23/7

Therefore, the value of k that makes the statement true is k = 23/7.

Answer 2

Explanation:

m1=y2-y1/x2-x1=12-5/k-3=7/k-3

m2=10-7/2-0=3/2

two lines with slopes m1 and m2 are perpendicular if m1×m2 = -1.

(7/k-3) *3/2=-1

3(7)=-2(k-3)

21=-2k+6

-2k=15

k=-15/2