Question

Find the equations of the tangent and the normal to the curve $x^2+y^2=5$, where the tangent is parallel to the line $2x-y+1=0$.

Answer 1

$x^2+y^2=5$

The above equation represents a circle centred at (0,0) and

radius $=\sqrt{5}$

$\Rightarrow$ The equation of tangent at a pt $(x_1,y_1)$ on the circle is

$x{x}_1+y{y}_1=5$

$\because$ The tangent is parallel to $2x-y+1=0$

$\Rightarrow \frac{x_1}{2}=\frac{y_1}{-1}$

$\because x_1^2+y_1^2=5$

$\Rightarrow 4_1^2+y_1^2=5$

$\Rightarrow y_1=\pm 1$

$\Rightarrow x_1=\pm 2$

$\Rightarrow$ When $y=1,x=-2\to$ not acceptable

$y=-1,x=2$

$\Rightarrow$ The tangents is $2x-y=5$

$\therefore$ The equation of normal is $y-0=\frac{-1-0}{2-0}(x-0)$

$\Rightarrow 2y+x=0\to$ normal

$2x-y=5\to$ tangent