The equation of the tangent to the circle, given by x = 5 \c... | Mathematics - Circle and Tangents | SolveeIt

The equation of the tangent to the circle, given by $x = 5 \cos \theta, y = 5 \sin \theta$ at the point $\theta = \frac{\pi}{3}$ on it, is

$x - \sqrt{3}y = -5$

$x + \sqrt{3}y = 10$

$\sqrt{3}x + y = 5\sqrt{3}$

$\sqrt{3}x - y = 0$

Answer

x + √3y = 10

Explanation

For a circle with parametric equations

$x = 5\cos\theta, y = 5\sin\theta$ ,

the equation of the tangent at the point corresponding to $\theta$ is given by:

$x\cos\theta + y\sin\theta = 5$ .

At $\theta = \frac{\pi}{3}$ , we have:

$\cos\frac{\pi}{3} = \frac{1}{2}, \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$ .

Substitute these into the tangent equation:

$x\left(\frac{1}{2}\right) + y\left(\frac{\sqrt{3}}{2}\right) = 5$ .

Multiply through by 2 to eliminate fractions:

$x + \sqrt{3} y = 10$ .

This corresponds to Option B.

Question: The equation of the tangent to the circle, given by $x = 5 \cos \theta, y = 5 \sin \theta$ at the po...