Question

Find equation of tangent to curve y=x-7/(x-2)(x-3)

Answer 1

The equation of tangent to curve y=x-7/(x-2)(x-3) is given by,

Given,

Equation of the curve y = x-7/(x-2)(x-3)

Let the curve cuts the x-axis at (x, 0)

⇒ 0 = x-7/(x-2)(x-3)

⇒ x - 7 = 0

∴ x = 7

The point is given by, (7, 0)

Differentiating the given equation of curve, we get,

$\dfrac{d}{dx} (y) = \dfrac{d}{dx}\left(\dfrac{x-7}{\left(x-2\right)\left(x-3\right)}\right)\\\\\\=\dfrac{\frac{d}{dx}\left(x-7\right)\left(x-2\right)\left(x-3\right)-\frac{d}{dx}\left(\left(x-2\right)\left(x-3\right)\right)\left(x-7\right)}{\left(\left(x-2\right)\left(x-3\right)\right)^2}\\\\\\=\dfrac{1\cdot \left(x-2\right)\left(x-3\right)-\left(2x-5\right)\left(x-7\right)}{\left(\left(x-2\right)\left(x-3\right)\right)^2}\\\\\\=\dfrac{-x^2+14x-29}{\left(x-2\right)^2\left(x-3\right)^2}$

$\dfrac{dy}{dx}_{(7, 0)} = \dfrac{-7^2+14(7)-29}{\left(7-2\right)^2\left(7-3\right)^2}$

= 20/400 = 1/20

Therefore the equation of tangent,

y - y1 = m (x - x1)

y - 0 = 1/20 (x - 7)

20 y = x - 7

∴ x - 20y - 7 = 0