A-Level Edexcel Maths Pure Modelling with Functions: The circle C has radius 5 and to

The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4 - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 4

Question 4

The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4. (a) Write down an equation for the circle C, that is shown in Figure 4. ... show full transcript

Worked Solution & Example Answer:The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4 - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 4

Step 1

Write down an equation for the circle C, that is shown in Figure 4.

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Answer

To find the equation of the circle C, we note that it has a radius of 5 and touches the y-axis at (0, 9). Therefore, the center of the circle is at (5, 9).

The general form of the equation of a circle is:

$(x - h)^2 + (y - k)^2 = r^2$

Where ( (h, k) ) is the center and ( r ) is the radius. Plugging in our values:

$(x - 5)^2 + (y - 9)^2 = 5^2$

Thus, the equation of circle C is:

$(x - 5)^2 + (y - 9)^2 = 25$

Step 2

Find the length of PT.

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Answer

To find the length of PT where P is (8, -7) and T is the tangent point on circle C:

Find the coordinates of point T.
- Since PT is tangent to the circle at point T, the radius CT is perpendicular to PT. The center C is (5, 9).
- The slope of CT is given by: $m_{CT} = \frac{9 - (-7)}{5 - 8} = \frac{16}{-3} = -\frac{16}{3}$
- The slope of PT, being perpendicular, is the negative reciprocal: $m_{PT} = \frac{3}{16}$
- Using point-slope form from point P(8, -7): $y + 7 = \frac{3}{16}(x - 8)$
- Rearranging gives the equation of line PT.
Get the intersection point T with circle C.
- Substitute the y-value from the line equation into the circle's equation to solve for x.
Calculate the distance PT.
- Use the distance formula: $PT = \sqrt{ (8 - x_T)^2 + (-7 - y_T)^2 }$
- After finding coordinates of T, plug into the distance formula to get the final length.

The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4 - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 4

Worked Solution & Example Answer:The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4 - Edexcel - A-Level Maths Pure - Question 4 - 2013 - Paper 4

Write down an equation for the circle C, that is shown in Figure 4.

Find the length of PT.

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