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Mathematics

Assertion (A) : Given two straight lines 3x - 2y = 5 and 2x + ky + 7 = 0 are perpendicular to each other when k = 3.

Reason (R) : Let AB and CD be two mutually perpendicular lines and their inclinations be α and θ respectively, then tan θ = -cot α.

  1. A is true, R is false.

  2. A is false, R is true.

  3. Both A and R are true.

  4. Both A and R are false.

Straight Line Eq

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Answer

1st equation : 3x - 2y = 5

⇒ 2y = 3x - 5

⇒ y = 3x−52\dfrac{3x - 5}{2}

⇒ y = 32x−52\dfrac{3}{2}x - \dfrac{5}{2}

Comparing above equation with y = mx + c, we get :

Slope of first line = 32\dfrac{3}{2}

2nd equation : 2x + ky + 7 = 0

⇒ ky = -2x - 7

⇒ y = −2x−7k\dfrac{-2x - 7}{k}

⇒ y = −2kx−7k\dfrac{-2}{k}x - \dfrac{7}{k}

Comparing above equation with y = mx + c, we get :

Slope of second line = −2k\dfrac{-2}{k}

We know that,

Product of slope of perpendicular lines = -1.

⇒ Slope of first line × Slope of second line = -1

⇒ 32×−2k=−1\dfrac{3}{2} \times \dfrac{-2}{k} = -1

⇒ −3k=−1\dfrac{-3}{k} = -1

⇒ k=−3−1k = \dfrac{-3}{-1} = 3.

∴ Assertion (A) is true.

Given, inclination of AB and CD are α and θ respectively.

Slope of AB = tan α

Slope of CD = tan θ

Since, AB and CD are perpendicular.

∴ Slope of AB × Slope of CD = -1

⇒ tan α × tan θ = -1

⇒ tan θ = −1tan α-\dfrac{1}{\text{tan α}}

⇒ tan θ = -cot α.

∴ Reason (R) is true.

Hence, Option 3 is the correct option.

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