SOLUTION: (1) A cirlce is defined by the equation x^2-6x+y^2-2y=6. (a) Rewrite the equation of the circle in the form:(x-a)^2+(y-b)^2=r^2 (b) Write down the coordinates of M,the centre

Question 1140775: (1) A cirlce is defined by the equation x^2-6x+y^2-2y=6.
(a) Rewrite the equation of the circle in the form:(x-a)^2+(y-b)^2=r^2
(b) Write down the coordinates of M,the centre of the circle.
(c) A tangent PQ is drawn from the point P(6;-2) to the circle, with Q the point of contact.Calculate the length of the tangent PQ.
Please can someone please help me with the question I begg
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!
(1) A cirlce is defined by the equation .
(a) Rewrite the equation of the circle in the form:

............complete square
....... for part is and for part is

(b) Write down the coordinates of M,the centre of the circle.
since and ,
the coordinates of M are: (,)=(,)

(c) A tangent PQ is drawn from the point P(;) to the circle, with the point of contact.Calculate the length of the tangent .
if is a tangent, it is perpendicular to the radius
points ,, and form right triangle where is hypotenuse, and are legs
we can find the length of using points P(;) and M(,)

->one leg
and
, the other leg will be:
P(;) and Q(,)

then we can find hypotenuse :
(,) and (,)

using Pythagorean theorem:

.............=>so point (,)

then substitute in Pythagorean theorem:

........solve for

and, then
=>
so (,)

the length of the tangent is: distance between
(;)
(,)

double check:

tangent is a line passing through points P(;) and (,), so find equation
first find a slope:

and use slope point formula

-> tangent line
line that passes through and is perpendicular to tangent and it is:

and use slope point formula

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The answer to part (c) from the other tutor is not for the question that is asked....

Complete the square in x and y to put the equation in standard form

The center is (3,1); the radius is 4.

We are done with parts (a) and (b) (done correctly by the other tutor).

(c) A tangent PQ is drawn from the point P(6,-2) to the circle, with Q the point of contact.

Consider the line determined by M and Q. Let the diameter of the circle contained in that line be AB, with B between M and Q. Then the rule relating the lengths of a tangent and a secant to a circle tell us

MA and MB are radii of the circle; length 4. MP by the Pythagorean Theorem is 3*sqrt(3). Then

ANSWER: PQ = sqrt(11)
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