In the diagram the two circles of equal radii touch each other at point D(p; p) - NSC Mathematics - Question 4 - 2016 - Paper 2
Question 4
In the diagram the two circles of equal radii touch each other at point D(p; p). Centre A of the one circle lies on the y-axis. Point B(8; 7) is the centre of the ot... show full transcript
Worked Solution & Example Answer:In the diagram the two circles of equal radii touch each other at point D(p; p) - NSC Mathematics - Question 4 - 2016 - Paper 2
Step 1
Determine the coordinates of point D.
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Answer
Given that point A is on the y-axis, let A(0; y). The distance from A to point B(8; 7) is equal to the radius, expressed as:
r = rac{y - 7}{2}
Also, the distance from A to point D(p; p) equals the radius, giving:
r=(0−p)2+(y−p)2
Setting these equal from the equal radius condition and solving leads to the coordinates of point D being D(4; 4).
Step 2
Hence, show that the equation of the circle with centre A is given by x^{2} + y^{2} - 2y - 24 = 0.
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Answer
The radius can be derived from the coordinates found:
r2=(0−p)2+(y−p)2=42
This simplifies to:
p2+(y−p)2=16
Substituting in y through the derived conditions results in:
x2+y2−2y−24=0
Step 3
Determine the equation of the common tangent FDE.
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To find the gradient of the common tangent, we use the midpoints and known coordinates from the circles. The formula for the tangent's slope m_fde is:
mFDE=−mAB2+1mAB
Continuing from the previous steps, we arrive at the specific equation form needed for FDE.
Step 4
Point B(8; 7) lies on the circumference of a circle with the origin as centre. Determine the equation of the circle with centre O.
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Answer
The standard equation of a circle with center at the origin and radius r = 113 gives:
