Question

Find the equation of the tangent at the origin to the circle ${x^2} + {y^2}$ - 4$x$ - 10$y$ = 0.

Answer 1

Hint: Differentiate the given equation of circle wrt $x$ to find the expression of $\dfrac{{dy}}{{dx}}$, This will be the slope of the tangent to the circle, find its value by substituting the point through which it is passing, then at last use point-slope form of the equation of line to get the answer.

Complete step-by-step answer:

The equation given to us by the question is ${x^2} + {y^2} - 4x - 10y = 0$. We will differentiate both the sides with respect to x in order to solve this further. We get-
$
   \Rightarrow {x^2} + {y^2} - 4x - 10y = 0 \\
    \\
   \Rightarrow \dfrac{{d}}{{dx}}\left( {{x^2} + {y^2} - 4x - 10y} \right) = \dfrac{{d}}{{dx}}0 \\
    \\
   \Rightarrow 2x + 2y.\dfrac{{dy}}{{dx}} - 4 - 10\dfrac{{dy}}{{dx}} = 0 \\
    \\
   \Rightarrow \dfrac{{dy}}{{dx}}\left( {2y - 10} \right) = 4 - 2x \\
    \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{4 - 2x}}{{2y - 10}} \\
$
As we know that the value of x and y is $(0,0)$, putting this value in the above equation, we get-
$
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{4 - 2x}}{{2y - 10}} \\
    \\
   \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{4}{{ - 10}} = - \dfrac{2}{5} \\
$
Now, to find out the equation of the tangent, we will-
We know that the value of ${x_1}$ and ${y_1}$is $(0,0)$ and the value of $\dfrac{{dy}}{{dx}}$ is $ - \dfrac{2}{5}$. So,
We also know that $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$, where m is the slope i.e. $\dfrac{{dy}}{{dx}}$.
Putting all the values in the above formula, we will get the equation of the tangent-
$
   \Rightarrow \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right) \\
    \\
   \Rightarrow \left( {y - 0} \right) = - \dfrac{2}{5}\left( {x - 0} \right) \\
    \\
   \Rightarrow y = - \dfrac{2}{5}x \\
    \\
   \Rightarrow 2x + 5y = 0 \\
$
Hence, the required equation of the tangent is $2x + 5y = 0$.

Note: In such questions where you must find the equation of a tangent, do not forget the formula of finding the equation i.e. $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$. Where, m is the slope which you will get by differentiating the given equation of the circle.