How to find a tangent to the curve $y = 12/x + 2$ without using differentiation at all

Question

Please can you explain how to solve this without using differentiation at all.

The straight line $y = mx + 14$ is a tangent to the curve $y = \frac{12}{x} + 2$ at the point $P$. Find the value of the constant $m$ and the coordinates of $P$.

Answer 1

Hint. Solve the equation $mx + 14=12/x + 2$. Find the value of $m$ such that the (quadratic) equation has just one solution, that is when its discriminant is zero. Then the unique solution is the $x$-coordinate of the point of tangency $P$.

Answer 2

WLOG $P\left(\dfrac1a,12a+2\right)$

$$\dfrac{dy}{dx}_{(\text{at }x=1/a)}=-\dfrac{12}{x^2}_{(\text{at }x=1/a)}=-12a^2$$

$$\dfrac{y-(12a+2)}{x-\dfrac1a}=-12a^2\iff12a^2 x+y-24a-2=0$$

This will be identical with $$mx-y+14=0$$

$$\iff\dfrac{12a^2}m=\dfrac1{-1}=-\dfrac{24a+2}{14}$$

Can you take it from here?