Find the equation of the tangent to the curve $y = \sqrt{3 x - 2}$ which is parallel to the line 4x – 2y + 5 = 0

Solution

$Curve y = \sqrt{3 x - 2} \frac{dy}{dx} = \frac{1}{2} {(3 x - 2)}^{\frac{- 1}{2}} x 3 \Rightarrow \frac{dy}{dx} = \frac{3}{2 \sqrt{(3 x - 2)}} . . . . . . . . . (i)$

Since, the tangent is parallel to the line 4x - 2y = - 5

Therefore, slope of tangent can be obtained from equation

$y = \frac{4 x}{2} + \frac{5}{2} Slope = 2 \Rightarrow \frac{dy}{dx} = 2 . . . . . . . . . . . (ii)$

Comparing equations (i) and (ii), we have,

$\frac{3}{2} x \frac{1}{\sqrt{3 x - 2}} = 2 \Rightarrow \frac{1}{\sqrt{3 x - 2}} = \frac{4}{3} \Rightarrow \frac{1}{3 x - 2} = \frac{16}{9} \Rightarrow 9 = 48 x - 32 \Rightarrow x = \frac{41}{48} We have, y = \sqrt{3 x - 2}$

Thus, substituting the value of x in the above euation,

$y = \sqrt{3 x \frac{41}{48} - 2} \Rightarrow y = \sqrt{\frac{41}{16} - 2} \Rightarrow y = \sqrt{\frac{41 - 32}{16}} \Rightarrow y = \sqrt{\frac{9}{16}} \Rightarrow y = \frac{3}{4}$

Equation of tangent is

$(y - \frac{3}{4}) = 2 (x - \frac{41}{48}) \Rightarrow (y - \frac{3}{4}) = 2 x - \frac{41}{24} \Rightarrow y = 2 x - \frac{41}{24} + \frac{3}{4} \Rightarrow y = 2 x - \frac{41}{24} + \frac{18}{24} \Rightarrow y = 2 x - \frac{23}{24} \Rightarrow 24 y = 48 x - 23 \Rightarrow 48 x - 24 y - 23 = 0$

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Application Of Derivatives

Find the equation of the tangent to the curve $y = \sqrt{3 x - 2}$ which is parallel to the line 4x – 2y + 5 = 0

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

Find the equation of the tangent to the curve y = 3x - 2 which is parallel to the line 4x – 2y + 5 = 0

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the equation of the tangent to the curve $y = \sqrt{3 x - 2}$ which is parallel to the line 4x – 2y + 5 = 0