Question

A circle has centre O. The point P(2,4) lies on the circumference of the circle. What is the gradient of the tangent to the circle at P? Give your answer as an exact whole number or decimal, or as a fraction in the form ( a)/(b).

Answer 1

The gradient of the tangent to the circle at point P is - 0.5.

The gradient (slope) of the tangent to a circle at a given point is perpendicular to the radius at that point. The radius of the circle is the line segment from the center (O) to the point on the circumference (P).

If we denote the center of the circle as O(0, 0) and the point on the circumference as P(2, 4), the gradient of the radius OP can be calculated as:

`\text { Gradient of } O P=\frac{\text { change in } y}{\text { change in } x}=(4-0)/(2-0)=(4)/(2)=2`

The gradient of the tangent at point P is the negative reciprocal of the gradient of OP. Therefore, the gradient of the tangent is:

`\text { Gradient of tangent at } \mathrm{P}=-\frac{1}{\text { Gradient of } \mathrm{OP}}=-(1)/(2)` (-0.5 in decimal).

Answer 2

The gradient of a tangent line to a circle at a given point is equal to the negative reciprocal of the gradient of the radius of the circle passing through that point.

Step-by-step explanation:

The gradient of a tangent line to a circle at a given point is equal to the negative reciprocal of the gradient of the radius of the circle passing through that point. To find the gradient of the tangent at point P(2,4), we need to find the gradient of the radius at that point. The radius is a line connecting the center of the circle O to point P. The center of the circle is not given, so we cannot find the exact gradient of the radius. However, we can still find the gradient of the tangent using the concept of negative reciprocal. The negative reciprocal of the gradient of the radius will be the gradient of the tangent line.

Let's assume the center of the circle is at (a,b). The gradient of the radius from center O to point P is given by:

mOP = (4 - b)/(2 - a)

The gradient of the tangent line will be the negative reciprocal of mOP:

mtangent = -1/mOP