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Find the length of the radius of the circle c, and hence write down the equation of c - Leaving Cert Mathematics - Question iv - 2021

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Find the length of the radius of the circle c, and hence write down the equation of c. Radius: _____________ Equation: _____________

Worked Solution & Example Answer:Find the length of the radius of the circle c, and hence write down the equation of c - Leaving Cert Mathematics - Question iv - 2021

Step 1

Find the length of the radius of the circle c

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Answer

To find the radius of the circle, we can use the standard form of a circle's equation, which is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2 where (h,k)(h, k) is the center and rr is the radius.

Given the center (2, -1) and the equation of the circle based on the marking scheme's information, we can substitute these into the formula:

(x−2)2+(y+1)2=52(x - 2)^2 + (y + 1)^2 = 5^2

This gives us: (x−2)2+(y+1)2=25(x - 2)^2 + (y + 1)^2 = 25

Therefore, the radius is: r=5r = 5

Step 2

Write down the equation of c

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Answer

Using the derived information, the equation of the circle c is:

(x−2)2+(y+1)2=25(x - 2)^2 + (y + 1)^2 = 25

Step 3

The point P(2, k) is in the first quadrant and is on c.

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Answer

Since the point P(2, k) lies on the circle, we substitute (x,y)=(2,k)(x, y) = (2, k) into the circle's equation:

(2−2)2+(k+1)2=25(2 - 2)^2 + (k + 1)^2 = 25

This simplifies to: (k+1)2=25(k + 1)^2 = 25

Taking the square root gives: k+1=5extork+1=−5k + 1 = 5 ext{ or } k + 1 = -5

From the first equation: k=5−1=4k = 5 - 1 = 4

From the second equation: k=−5−1=−6k = -5 - 1 = -6

Since we require kk to be in the first quadrant, we take: k=4k = 4

Step 4

Plot the point P(2, k)

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Answer

The coordinates of P are (2, 4). This point should be located in the first quadrant on the diagram provided, confirming that it lies on the circle.

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