Thread: points of tangency (plz help)

1. points of tangency (plz help)

1.)for the circle x^2+y^2+6x-4y+3=0
find the equation of the tangent line at (-2,5)

2.)a circle is tangent to the y-axis at y=3 and has one x-intercept at x=1
determine the other x-int and deduce the equation of the circle

any help or hints on how to solve this problem would be GREATLY appreciated

2. Originally Posted by saar4ever
1.)for the circle x^2+y^2+6x-4y+3=0
find the equation of the tangent line at (-2,5)

2.)a circle is tangent to the y-axis at y=3 and has one x-intercept at x=1
determine the other x-int and deduce the equation of the circle

any help or hints on how to solve this problem would be GREATLY appreciated

1) Using implicit differentiation,

$2x+2y\frac{dy}{dx}+6-4\frac{dy}{dx}=0$ -------- (1)

$2(-2)+2(5)\frac{dy}{dx}+6-4\frac{dy}{dx}=0$

$-4+10\frac{dy}{dx}+6-4\frac{dy}{dx}=0$

$6\frac{dy}{dx}=-2$

$\frac{dy}{dx}=-\frac{1}{3}$

3. thanks so much alexmahone but im still stumped at the second problem
btw i did it a bit diff way
i found the slope of (-2,5) to (-3,2)<-- Center
and found out it was m=3
so reciprical of that is -1/3
and then i plugged it into the point slope form and got
y-5=-1/3(x+2)
so y=-x/3+13/3
right?

4. here is the solution

let the coordinate of other intercept be (a,0) (its on x axis)
the coordinate of middle point of (1,0) and (a,0) will be ((a+1)/2,0)(let us name it as A).its magnitude will be equal to the length of radius.
a line drawn perpendicular to this middle point and y intercept will intersect at centre of the circle (let centre of circle be O).
now it can be easily observed that coordinate of centre of circle will be ((a+1)/2,3)
let intercept (1,0) be B
so triangle AOB is a right angle triangle where AO=3 (length of y intercept),OB=length of radius=(a+1)/2,AB=[(a+1)/2]-1=(a-1)/2
now applying pythagores theorem we have
[(a+1)/2]^2=[(a-1)/2]^2+9
solving this we get a=9
so the coordinate of other intercept will be (9,0) and radius will be 5 units snd coordinate of centre will be (5,3)
and equation of circle will be
(x-5)^2+(y-3)^2=25 or
x^2+y^2-10x-6y+9=0

5. Thank you soooo much for helping me on the second problem, nikhil.
You are the best
I really like the way u solved it step by step
thx so much