Given: two curves y² = 4x and x² + y² – 6x + 1 = 0

To prove: the two curves touch each other at point (1,2)

Explanation:

Now given x² + y² – 6x + 1 = 0

Differentiating this with respect to x, we get

Applying sum rule of differentiation, we get

Solving the above equation at point (1,2), we get

Also given y² = 4x

Differentiating this with respect to x, we get

Now applying the product rule of differentiation, we get

Solving the above equation at point (1,2), we get

From equation (i) and (ii),

∴ m₁ = m₂

Therefore, both curves touch each other at point (1,2).

Hence proved