The equation of the line touching both the parabola y² = 4x and ${\mathrm{x}}^2=-32\mathrm{y}\text{ is }\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$. Then the value of a + b +c is

The equation of tangent in terms of the slope of the parabola y² = 4x is $\mathrm{y}=\mathrm{mx}+\frac{1}{\mathrm{m}}$
Since (i) is also the tangent of x² = - 32 !, wa have
$\begin{array}{r}{\mathrm{x}}^2=-32\left(\mathrm{mx}+\frac{1}{\mathrm{m}}\right)\\ \mathrm{or} {\mathrm{x}}^2+32\mathrm{mx}+\frac{32}{\mathrm{m}}=0\end{array}$
The above equation must have equal roots.
Hence, its discriminant must be zero. Therefore,

$\begin{array}{r}(32\mathrm{m}{)}^2=4\times 1\times \frac{32}{\mathrm{m}}\\ \mathrm{i}.\mathrm{e}.,{\mathrm{m}}^3=\frac{1}{8}\text{ or }\mathrm{m}=\frac{1}{2}\end{array}$

From (i), $\mathrm{y}=\frac{\mathrm{x}}{2}+2\text{ or }\mathrm{x}-2\mathrm{y}+4=0$