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The equation of a curve is $y = x^2$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 3

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The equation of a curve is $y = x^2$. A is the point where the curve intersects the y-axis. (a) State the coordinates of A. The equation of circle C is $x^2 + y^2 ... show full transcript

Worked Solution & Example Answer:The equation of a curve is $y = x^2$ - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 3

Step 1

State the coordinates of A.

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Answer

To find the coordinates of point A, we evaluate the curve at the point where it intersects the y-axis. At the y-axis, the value of x is 0.

Substituting into the equation:

y=(0)2=0y = (0)^2 = 0

Thus, the coordinates of A are (0, 0).

Step 2

Label with coordinates the centre of circle B.

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Answer

The original circle C has its centre at the origin (0, 0) with a radius of 4 (since (x^2 + y^2 = 16) implies a radius of (r = 4)). After translating by the vector (egin{pmatrix} 3 \ 0 \ 0 \end{pmatrix}), the new centre for circle B becomes (3, 0).

  • Centre of Circle B: (3, 0)

Step 3

Label with coordinates any points of intersection with the x-axis.

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Answer

To find the points of intersection of circle B with the x-axis, we set (y = 0) in the equation of circle B:

The equation of circle B, after the translation, is:

(x−3)2+y2=16(x - 3)^2 + y^2 = 16

Setting (y = 0):

(x−3)2=16(x - 3)^2 = 16

Taking square roots gives:

x−3=4orx−3=−4x - 3 = 4 \quad \text{or} \quad x - 3 = -4

Thus,

  • x=7  ⟹  (7,0)x = 7 \implies (7, 0)
  • x=−1  ⟹  (−1,0)x = -1 \implies (-1, 0)

The points of intersection with the x-axis are (7, 0) and (-1, 0).

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