Question

$Find \: \: the \: \: point \: \: at \: \: which \: \: the \: \: tangent \: \: to \: \: the \: \: curve \: \: y \: \: = \sqrt{4x - 3} - 1 \: \: has \: \: its \: \: slope \: \: \frac{2}{3}$
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Answer 1

Answer:

$\textsf{Given slope of the tangent to the curv is=} \frac{2}{3}$

we know that

$\textsf{slope of tangent=} \frac{dy}{dx}$

$\frac{2}{3}= \frac{dy}{dx}$

$\frac{dy}{dx}= \frac{2}{3}$

$\frac{d( \sqrt{4x - 3 - 1)} }{dx} = \frac{2}{3}$

$\frac{1}{2 \sqrt{4x - 3}} \times 4 - 0 = \frac{2}{3}$

$\frac{2}{\sqrt{4x-3} }= \frac{2}{3}$

$3=\sqrt{4x-3}$

$\sqrt{4x-3}=3$

$\textsf{squaring both sides}$

$4x-3=9$

$4x=12$

$x=3$

$\textsf{finding y for x=3}$

$y=\sqrt{4x-3-1}$

$\sqrt{12-3-1}$

$y=\sqrt{9} -1$

$=3-1$

$=2$

$\textsf{Hence Required points is (3,2)}$

Answer 2

EXPLANATION.

Tangent to the curve y = (√4x - 3) - 1.

Slope = 2/3.

As we know that,

Tangent of the curve = Slope.

⇒ dy/dx = 1/2√4x - 3 x 4.

⇒ dy/dx = 2/√4x - 3.

⇒ 2/√4x - 3 = 2/3.

⇒ √4x - 3 = 3.

Squaring on both side of the equation, we get.

⇒ (√4x - 3)² = (3)².

⇒ 4x - 3 = 9.

⇒ 4x = 9 + 3.

⇒ 4x = 12.

⇒ x = 3.

Put the value of x = 3 in the equation, we get.

⇒ y = (√4(3) - 3) - 1.

⇒ y = (√12 - 3) - 1.

⇒ y = (√9) - 1.

⇒ y = (3) - 1.

⇒ y = 2.

Point are = (3,2).

                                                                                                                         

MORE INFORMATION.

Equation of tangent.

Equation of tangent to the curve y = f(x) at p(x₁, y₁) is,

(1) = (y - y₁) = m(x - x₁).

(2) = The tangent at (x₁, y₁) is parallel to x-axes = (dy/dx) = 0.

(3) = The tangent at (x₁, y₁) is parallel to y-axes = (dy/dx) = ∞.

(4) = The tangent lines makes equal angles with the axes = (dy/dx) = ± 1.