Find the equation of tangent to the curve , at the point, where it cuts the x-axis.

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.