Equation of a Circle

Edexcel IAL WMA12/01/P2 /June 2019/Q2 (Equation of Circles)

A circle C has equation

x2+y2+4x-10y-21=0

Find​​ 

•  ​​​​ 

• The coordinates of the center of C

• The exact value of the radius of C.

(3)

The point P(5, 4) lies on C.

• Find the equation of the tangent to C at P, writing your answer in the form​​ y=mx+c,​​ where m and c are constants to be found.

(4)

SOLUTION​​ 

a- i-​​ Changing the equation to the standard circle equation.

x2+4x+y2-10y=21

Completing the square for​​ x​​ terms and​​ y​​ terms.​​ 

x+22-4+y-52-25=21

x+22+y-52=50

Deducing the centre of the circle from the equation of the circle.​​ 

On comparing​​ x+22+y-52=50​​ with​​ x-a2+y-b2=r2, the centre of the circle comes out to be:

x-a2=x+22

a=-2

And​​ 

y-b2=y-52

b=5

Hence, the centre is​​ -2, 5

a- ii-​​ 

On comparing​​ x+22+y-52=50​​ with​​ x-a2+y-b2=r2, the radius of the circle comes out to be:

r2=50

r=50

r=52

b-​​ 

So, first finding the gradient of normal or​​ mOP, where O (-2, 5) and P (5, 4).

mOP=y2-y1x2-x1

mOP=4-55+2

mOP= -17

Since tangent and normal are perpendicular to each other.

mN×mT=-1

mOP×mT=-1

mT= 7

Now, using the point-slope formula. Substituting the gradient ​​ and point P as the tangent passes through it.

y-y1=mx-x1

y-y1=mT(x-x1)

y-4=7 x-5

y-4=7x-35

y=7x-31