Maths a level Q

Im not sure about the following Q:
The line 𝑥+5𝑦=𝑘 is a tangent to the curve 𝑥$−4𝑦=10. Find the value of the constant 𝑘.
I substituted the equation of the line into the curve, and I got
k^2-10yk+25y^2-4y-10=0
I wanted to use the discriminant but I am struggling to find the values of a, b and c

"a" is the coefficient of $y^2$. This is just 25.

"b" is the coefficient of $y$. But there is -10yk and -4y terms; just combine them into one! $(-10k-4)y$ therefore the coefficient is just $-10k-4$.

"a" is the coefficient of $y^2$. This is just 25.

"b" is the coefficient of $y$. But there is -10yk and -4y terms; just combine them into one! $(-10k-4)y$ therefore the coefficient is just $-10k-4$.

Im not sure if i have made a mistake I got that k=123/10