# State whether the following statements are true or false. Justify your answer.A circle has its centre at the origin and a point $P(5,0)$ lies on it. The point $\mathrm{Q}(6,8)$ lies outside the circle.

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Given:

A circle has its centre at the origin and a point $P(5,0)$ lies on it. The point $\mathrm{Q}(6,8)$ lies outside the circle.

To do:

We have to find whether the given statement is true or false.

Solution:

The distance between origin $O(0,0)$ and $P(5,0)$ is,

$OP=\sqrt{(5-0)^{2}+(0-0)^{2}}$

$=\sqrt{5^{2}}$

$=5$

This implies,

The radius of the circle is 5 units.

If the point $\mathrm{Q}(6,8)$ lies outside the circle, then the distance between the origin and point $Q$ is greater than the radius of the circle.

The distance between $O(0,0)$ and $Q(6,8)$ is,
$OQ=\sqrt{(6-0)^{2}+(8-0)^{2}}$

$=\sqrt{6^{2}+8^{2}}$

$=\sqrt{36+64}$

$=\sqrt{100}$

$=10$

$OQ=10>5$

The distance between $O(0,0)$ and $Q(6,8)$ is greater than the radius of the circle.

Hence, the point $\mathrm{Q}(6,8)$ lies outside the circle.

Updated on 10-Oct-2022 13:28:28