If the tangents on the ellipse 4 x ^ 2 + y ^ 2 = 8 at the points...

$4{x}^{2} + {y}^{2} = 8$
$⇒ 8x + 2y\dfrac{dy}{dx} = 0$
$\dfrac{dy}{dx} = - \dfrac{8x}{2y} = - \dfrac{4x}{y}$
$∴$ Slope at \left(1, 2\right), {m}_{1} = - 2$<br /> Slope at \left(a, b\right), {m}_{2} = - \dfrac{4a}{b}$
$∴ - 2 \times - \dfrac{4a}{b} = - 1$
$b = - 8a$
Again,
$4{a}^{2} + {b}^{2} = 8$
$⇒ 4{a}^{2} + 64{a}^{2} = 8$
$⇒ 68{a}^{2} = 8$
$⇒ {a}^{2} = \dfrac{8}{68}$

$⇒ {a}^{2} = \dfrac{2}{17}$