Magnetic field at the focus of an ellipse!

Using Biot-Savart Law's, the magnetic field at the focus $F$ by an infinitesimal segment of the wire is $dB = \frac{\mu_{0}I}{4\pi} \frac{dl \sin(\theta+\alpha)}{r^{2}}$

Now, we have $\sin(\theta+\alpha) = \sin\theta\cos\alpha + \cos\theta\sin\alpha$ Since $\tan\alpha = -\frac{dy}{dx}$ , then we have $\sin(\theta+\alpha) = -\sin\theta\frac{dx}{dl} + \cos\theta\frac{dy}{dl}$

Using relation, $x = r(\theta)\cos\theta$ and $y = r(\theta)\sin\theta$ , where $r(\theta) = \frac{a(1-e^2)}{1 + e\cos\theta}$ is the equation of ellipse in polar form with $e = \sqrt{1- (b/a)^2}$ is an eccentricity of the ellipse, then we get $dl \sin(\theta+\alpha) = r(\theta)d\theta$

Then the magnetic field in the focus is given by $B = \frac{\mu_{0}Ia}{4\pi b^{2}} \int_{0}^{2\pi}\: (1+ e\cos\theta ) \:d\theta = \frac{\mu_{0}Ia}{2 b^{2}}$

Using the data, we have $B = 16\pi$ Tesla or $k=16$ , then $k^{2} = 256$ .