Question

If a line passing through p(2,3) intersects the line x+y=7 at a point 4 units from p find the slope of the line

Answer 1

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Answer 2

slope of line is -2

a line passing through p(2,3) intersects the line x + y = 7 at a point of 4 unit from P.

let slope of unknown line is m.

since line passing through (2,3)

equation of line, (y - 3) = m(x - 2)

⇒y = mx - 2m + 3

now intersection of x + y = 7 and y = mx -2m + 3

7 - x = mx - 2m + 3

4 + 2m = (m + 1)x

x = (4 + 2m)/(m + 1)

and y = 7 - x = (7m + 7 - 4 - 2m)/(m + 1)

= (5m + 3)/(m + 1)

now, distance between (2,3) and intersection is 4.

so, √[{(4m+2)/(m+1) - 2}² + {(5m+3)/(m+1) - 3}²] = 4

⇒4m² + 4m² = 16(m + 1)²

⇒8m² = 16m² + 32m + 16

⇒8m² + 32m + 16 = 0

⇒m² + 4m + 4 = 0

⇒m = -2

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