Question

Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is:
(A) (√5 - 1)/(√5 + 1)
(B) (1 - √5)/(1 + √5)
(C) (√7 - 1)/(√7 + 1)
(D) (1 - √7)/(1 + √7)

Answer 1

it is given that slope of a line passing through P(2, 3) and interesting the line x +y = 7 at a distance of 4 units from P.

let m is the slope of a line passing through P(2, 3).

angle between lines is θ [ see figure ]

[ distance of line x + y = 7 from point P(2,3) = |2 + 3 - 7|/√(1² + 1²) = √2.

so, base of triangle = √(4² - √2²) = √14

tanθ = √2/√14 = |m - 1|/|1 + m|

taking positive sign,

1/√7 = (m - 1)/(1 + m)

⇒1 + m = √7m - √7

⇒(1 + 7) = (√7 - 1)m

⇒m = (1 + √7)/(√7 - 1)

taking negative sign,

1/√7 = -(m - 1)/(1 + m)

⇒1 + m = -√7(m - 1)

⇒1 + m = -√7m + √7

⇒1 - √7 = -(1 + √7)m

⇒m = (√7 - 1)/(1 + √7)

so slope of unknown line passing through P(2,3) is (√7 - 1)/(1 + √7).

therefore option (C) is correct choice.

Answer 2

Step-by-step explanation:

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