Q165 · Practice questionTechnology-freeComplex unfamiliar6 marks
QUESTION 165 (6 marks)
A straight line has vector equation
A family of spheres has centre and radius . The diagram is schematic.
a)[3 marks]
For , determine the coordinates of the points where meets the sphere.
b)[3 marks]
Determine the radius for which the sphere is tangent to , and the coordinates of the tangency point.
WORKED SOLUTION
6 marksPractice marking scheme
ANSWER
(a) : intersections and . (b) Tangency radius ; point .
Worked solution
(a) The sphere equation is . Substitute and :
Thus or , giving the two points stated.
(b) For a general radius, the squared distance to the centre is
Its minimum is at . A sphere is tangent to the line when its radius equals this minimum distance, so and the tangency point is . The radius vector there is , whose dot product with the line direction is zero.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Substitute the line into the sphere equation.
Part a: Solve for both parameters 0 and 2/5.
Part a: Obtain both intersection points.
Part b: Complete the square or impose a repeated quadratic root.
Part b: Obtain radius sqrt(24/5).
Part b: Obtain the tangency point (1/5,2/5,2).
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