1. ## [Coordinate Geometry] Locus

A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

Thanks so much !

2. Originally Posted by CallMeBin
A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

Thanks so much !
Let C be (x, y)
AC is perpendicular to BC
So slope of AC*slope of BC = -1.
Find the slopes of AC and BC and find the locus of C.

3. So the slope/gradient will be in unknown as C(x,y) ?

4. Originally Posted by CallMeBin
A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

Thanks so much !
hi

OR

Use the phythagoras theorem . Let c be (x,y)

$(x+1)^2+(4-y)^2+(7-x)^2+(5-y)^2=65$

then simplify it , you will find the locus of c to be a circle because the point C can be any point on the circumference since the angle in the semicircle is a right angle .

5. Originally Posted by CallMeBin
A(-1,4) and B(7,5) are two ends of the diameter of a circle . Find the equation of the locus of point C such that <ACB is a right angle .

Thanks so much !
HI all
I think c will be any point belongs to the circle
(x-3)^2 + (y - 9/2)^2 = 65
except (-1 , 4) , (7 , 5 )
mrmohamed

hi

OR

Use the phythagoras theorem . Let c be (x,y)

$(x+1)^2+(4-y)^2+(7-x)^2+(5-y)^2=65$

then simplify it , you will find the locus of c to be a circle because the point C can be any point on the circumference since the angle in the semicircle is a right angle .
mrmohamed

7. Thanks . I've done it . But I still don't understand why will it become like that (My way)

May I ask why = 65 ?

8. Originally Posted by CallMeBin
Thanks . I've done it . But I still don't understand why will it become like that (My way)

May I ask why = 65 ?
65 is the length of the diameter calculated using the distance formula .

9. Originally Posted by CallMeBin
So the slope/gradient will be in unknown as C(x,y) ?
A(-1,4) and B(7,5)
Slope of AC = (y-4)/(x+1)
Slope of BC = (y-5)/(x-7)
They are perpendicular because ACB is a right angled triangle.So
(y-4)(y-5)/(x+1)(x-7) = -1.
After simplification you will get the locus of C.

10. Thanks everyone !

This question is so special . The questions given by my teacher all are given the ratio but this one use distance instead of ratio .