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AP EAMCET202119 Aug 2021Evening ShiftMathematicsApplication of DerivativesActual

Find the equation of a line passing through the point ( 4 , 3 ) , which cuts a triangle of minimum area from the first quadrant.

Options

  1. A3 x + 4 y = 24
  2. B2 x − y = 5
  3. C2 x + y = 8
  4. Dx − 2 y = 5

Correct answer

A. 3 x + 4 y = 24

Step-by-step solution

The equation of the straight line passing through the point 4 ,   3 , whose slope m (assume). y - 3 = m x - 4   ⇒   y = m x - 4 m + 3 So, x - intercept of the line is 4 - 3 m and y - intercept is 3 - 4 m , So, area of the given triangle is A = 1 2 × 4 - 3 m 3 - 4 m = 1 2 24 - 16 m - 9 m = 12 - 8 m - 9 2 m To find the min/max, d A d m = 0     ⇒   - 8 + 9 2 m 2   = 0   ⇒ m = ± 3 4 Second derivative test : d 2 A d m 2 m = - 9 m 3 ⇒ d 2 A d m 2 m

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