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The point P has coordinates (4,4) - Scottish Highers Maths - Question 16 - 2019
Question 16
The point P has coordinates (4,4). C is the centre of the circle with equation {x−1}^2 + {y+2}^2 = 25. (a) Show that the distance between the points P and C is giv... show full transcript
Worked Solution & Example Answer:The point P has coordinates (4,4) - Scottish Highers Maths - Question 16 - 2019
Step 1
Show that the distance between the points P and C is given by \(\sqrt{(4−1)^2 + (4−(−2))^2}\)
Answer
To calculate the distance between point P(4, 4) and center C(1, -2), we apply the distance formula:
Substituting the coordinates of P and C:
Calculating the components:
- For the x-coordinates: ( (4 - 1)^2 = 3^2 = 9 )
- For the y-coordinates: ( (4 - (-2))^2 = (4 + 2)^2 = 6^2 = 36 )
Thus,
Step 2
Hence, or otherwise, find the range of values of k such that P lies outside the circle.
Answer
The radius of the circle from the equation ( (x-1)^2 + (y+2)^2 = 25 ) is 5 (since ( r^2 = 25 )).
The distance from point P to center C is ( d = 3\sqrt{5} ).
To determine when point P lies outside the circle, we need:
Squaring both sides:
This inequality is always true. Now considering the value of k affecting the position, we need:
- Express ( k ) in terms of the inequality derived from the distance formula, say ( k < -\frac{1}{2} ) and ( k > 2 ).
Thus, the range of values of k such that P lies outside the circle is: