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# The differential equation of all straight lines touching the circle x^2+y^2=a^2 is

Samyak Jain
333 Points
2 years ago
Equation of tangent to the circle x2 + y2 = a2 in slope form is
y = mx $\dpi{80} \pm$ a$\dpi{80} \sqrt{1 + m^2}$      ...(1)
Differentiate (1) with respect to x. $\dpi{100} \Rightarrow$  dy/dx = m ,  $\dpi{80} \because$ a$\dpi{80} \sqrt{1 + m^2}$ is a constant, its differentiation is zero.
Put the value of m in (1).
So, y = (dy/dx)x $\dpi{80} \pm$ a$\dpi{80} \sqrt{1 + (dy/dx)^2}$   $\dpi{100} \Rightarrow$  y – x(dy/dx) = $\dpi{80} \pm$ a$\dpi{80} \sqrt{1 + (dy/dx)^2}$
Squaring both sides we get,
(y – x dy/dx)2 = a2 (1 + (dy/dx)2).
Option (b) is correct.