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In this lesson, we will learn how to find the equation of a sphere given the center and how to find the center and the radius given the sphere's equation.

Q1:

Find the equation of a sphere that passes through the points π΄ ( 9 , 0 , 0 ) , π΅ ( 3 , 1 3 , 5 ) , and πΆ ( 1 1 , 0 , 1 0 ) , given that its centre lies on the π¦ π§ -plane.

Q2:

A sphere is tangent to the π₯ π¦ -plane, and has its centre on the π§ -axis at a distance of 35 length units from the π₯ π¦ -plane. What is the equation of the sphere?

Q3:

The spheres with equations π₯ + π¦ + π§ = 9 2 2 2 and ( π₯ β 4 ) + ( π¦ + 2 ) + ( π§ β 4 ) = 9 2 2 2 intersect in a circle. Find the equation of the plane in which this circle lies.

Q4:

Find the point(s) of intersection of the sphere ( π₯ β 3 ) + ( π¦ + 1 ) + ( π§ β 3 ) = 9 2 2 2 and the line π₯ = β 1 + 2 π‘ , π¦ = β 2 β 3 π‘ , π§ = 3 + π‘ .

Q5:

Determine if the given equation π₯ + π¦ β π§ + 1 2 π₯ + 2 π¦ β 4 π§ + 3 2 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

Q6:

Give the equation of the sphere of centre ( 1 1 , 8 , β 5 ) and radius 3 in standard form.

Q7:

The line π₯ + 9 β 1 0 = π¦ + 4 β 4 = π§ β 8 5 is tangent to the sphere ( π₯ β 7 ) + ( π¦ + 3 ) + ( π§ β 7 ) = π 2 2 2 2 . Find the sphereβs radius to the nearest hundredth.

Q8:

Determine if the given equation 2 π₯ + 2 π¦ + 2 π§ + 4 π₯ + 4 π¦ + 4 π§ β 4 4 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

Q9:

Which of the following does the equation β β β π β β β β π β ο» β 1 0 β π β 6 β π + 1 0 β π ο + 5 0 = 0 2 represent?

Q10:

It can be shown that any four noncoplanar points determine a sphere. Find the equation of the sphere that passes through the points ( 0 , 0 , 0 ) , ( 0 , 0 , 2 ) , ( 1 , β 4 , 3 ) and ( 0 , β 1 , 3 ) .

Q11:

Determine if the given equation π₯ + π¦ + π§ + 2 π₯ β 2 π¦ β 8 π§ + 1 9 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

Q12:

Which of the following is the equation of a sphere of the centre ( 8 , β 1 5 , 1 0 ) and passing through ( β 1 4 , 1 3 , β 1 4 ) ?

Q13:

Determine the equation of a sphere with centre ( 0 , 1 , 0 ) , given that it touches one of the coordinate planes.

Q14:

Given that a sphereβs equation is ( π₯ + 5 ) + ( π¦ β 1 2 ) + ( π§ β 2 ) β 2 8 9 = 0 2 2 2 , determine its centre and radius.

Q15:

A sphere of radius 2 is tangent to all three coordinate planes. Given that the coordinates of its centre are all positive, what is the equation of this sphere?

Q16:

Determine the surface area of the sphere of equation π₯ + π¦ + π§ β 1 4 4 4 = 0 2 2 2 , leaving your answer in terms of π .

Q17:

Determine if the given equation π₯ + π¦ + π§ β 4 π₯ β 6 π¦ β 1 0 π§ + 3 7 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

Q18:

Given π΄ ( 0 , 4 , 4 ) , and that π΄ π΅ is a diameter of the sphere ( π₯ + 2 ) + ( π¦ + 1 ) + ( π§ β 1 ) = 3 8 ο¨ ο¨ ο¨ , what is the point π΅ ?

Q19:

Find the equation of the sphere concentric with π₯ + π¦ + π§ + π₯ β 5 π¦ + 4 π§ = 3 2 2 2 , but with twice the radius.

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