Use the slope formula $m = \dfrac{y_2-y_1}{x_2-x_1}$ with the points $(3,4)$ and $(k,10)$ to express the slope in terms of $k$.

For two lines to be perpendicular, how are their slopes related? Write an equation involving the two slopes you found.

After you set the product of the slopes equal to $-1$, clear the fraction, combine like terms, and solve the resulting linear equation for $k$.

In Desmos, type m_l = k - 1 and m_m = 6/(k - 3) to define the slopes of the two lines in terms of $k$. Desmos will automatically create a slider for $k$.

In a new line, type m_l * m_m so Desmos shows the product of the two slopes. Move the $k$ slider to each answer choice (convert fractions to decimals as needed) and observe the value of m_l * m_m.

The correct option is the one where m_l * m_m equals $-1$. That $k$ value is the answer to the problem.

The equation of line $\ell$ is

This is in slope-intercept form $y=mx+b$, where the slope is the coefficient of $x$.

So, the slope of line $\ell$ is

Line $m$ passes through $(3,4)$ and $(k,10)$.

Use the slope formula $m = \dfrac{y_2-y_1}{x_2-x_1}$:

If two non-vertical lines are perpendicular, the product of their slopes is $-1$.

So set up the equation

Start with

Multiply both sides by $(k-3)$:

Distribute on both sides:

Add $k$ to both sides and add $6$ to both sides:

Divide both sides by $7$:

So the correct answer is $\boxed{\dfrac{9}{7}}$.