Video: Pack 4 • Paper 3 • Question 18

Pack 4 • Paper 3 • Question 18


Video Transcript

There are two parts to this question. The first part says we are given a constant 𝑐. 𝐴 is the point where the line 𝑦 equals two 𝑐 to the power of 𝑥 plus three crosses the 𝑦-axis. Find the coordinates of the point 𝐴.

Any point that crosses the 𝑦-axis has an 𝑥-coordinate equal to zero. This means that we can substitute 𝑥 equals zero into the equation 𝑦 equals two 𝑐 to the power of 𝑥 plus three. This gives us 𝑦 equals two 𝑐 to the power of zero plus three. Zero plus three is equal to three. Therefore, our 𝑦-coordinate is two 𝑐 cubed. The coordinates of the point 𝐴 where the line crosses the 𝑦-axis at zero, two 𝑐 cubed.

The second part of our question says the following.

A circle has the equation 𝑥 squared plus 𝑦 squared minus 49 equals zero. The vector zero, two translates circle 𝐶 to circle 𝐵. Sketch circle 𝐵. Clearly label the coordinates of the center of circle 𝐵 and any points where 𝐵 intersects the 𝑦-axis.

The equation 𝑥 squared plus 𝑦 squared minus 49 equals zero can be rewritten as 𝑥 squared plus 𝑦 squared equals 49. This is the equation of a circle with center zero, zero and radius root 49 which is equal to seven. The circle 𝐶, as shown in the diagram, has center zero, zero. It intersects the 𝑥-axis at seven, zero and minus seven, zero and intersects the 𝑦-axis at zero, seven and zero, minus seven.

A translation of vector zero, two will increase all the 𝑦-coordinates by two and will move the circle up the coordinate grid. The center of the circle will move from zero, zero to zero, two and the radius will remain equal to seven. Circle 𝐵 will, therefore, intersect the 𝑦-axis at zero, nine and zero, minus five with a center of zero, two.

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