Find the equation

To find the equation of the tangent to the curve, we first need to find the slope of the tangent to the curve.

Slope of the tangent to the curve y is given by .

We have,

Dividing numerator and denominator by (x – 2)(x – 3),

And we have, . So, replace the value by y in the above equation.

We know that the tangent cuts the x-axis. This means that,

y = 0

Take

⇒ (x – 2)(x – 3) × 0 = x – 7

⇒ x – 7 = 0

⇒ x = 7

We have thus got the point which cuts the x-axis, that is, (7, 0).

We need to find the slope of the tangent at point (7, 0). So,

Now, equation of the tangent at point (x, y) with slope m is given by,

y – y₁ = m(x – x₁)

Now, replace x₁ by 7, y₁ by 0 and m by 1/20.

⇒ 20y = x – 7

⇒ x – 20y – 7 = 0

Thus, equation of the tangent to the curve is x – 20y – 7 = 0.