Find the value of '$k$' for which the quadratic equation $x^2-kx+9=0$ has real and distinct roots.

Find the value of '$k$' for which the quadratic equation $k{x}^2+2x+1=0$ has real and distinct roots.

Find the least value of $k$, when $\left(4-k\right)x^2+\left(2k+4\right)x+\left(8k+1\right)$ is a perfect square.

If the equation $\left(1+m^2\right)x^2+2mcx+\left(c^2-a^2\right)=0$ has equal roots, prove that $c^2=a^2\left(1+m^2\right)$.

If the roots of the equation $a{x}^2+2bx+c=0$ and $b{x}^2-2\sqrt{ac}x+b=0$ are simultaneously real, prove that $b^2=ac$.

If the roots of the equation $\left(a^2+b^2\right)x^2-2\left(ac+bd\right)x+\left(c^2+d\right)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$.

Note: The question given in the textbook seems to be wrong. The correct question should be as below:

If the roots of the equation $\left(a^2+b^2\right)x^2-2\left(ac+bd\right)x+\left(c^2+d^2\right)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$.

If the equation $\left(1+m^2\right)x^2+2mcx+\left(c^2-a^2\right)=0$ has equal roots, prove that $c^2=a^2\left(1+m^2\right)$.