Maths Help GEOMETRY: A circle c is given parametrically by the equations ....

[asred9]

[h=1]Maths Help GEOMETRY: A circle c is given parametrically by the equations: x=√2 * cosθ - 2 & y=√2 sinθ +3 0 ≤ θ ≤ 2π?[/h]a) State the radius & the coordinates of the circle C. Hence express C in Cartesian form.
b) The straight line with equation y=-x+c is a tangent to the circle C. Find the possible values of c.
c) Find the equation of circle K which has the line joining the center of the circle C & the origin as its diameter.

I managed to find the radius which is √2 and the center which is (-2,3), I expressed C in cartesian form whihc i got (x+2)² + (y-3)²=2

Can anyone please help me in b & c. The answers together with how you ended up for that result are highly appreciated.

Thanks

 

[Steven G]

Maths Help GEOMETRY: A circle c is given parametrically by the equations: x=√2 * cosθ - 2 & y=√2 sinθ +3 0 ≤ θ ≤ 2π?

a) State the radius & the coordinates of the circle C. Hence express C in Cartesian form.
b) The straight line with equation y=-x+c is a tangent to the circle C. Find the possible values of c.
c) Find the equation of circle K which has the line joining the center of the circle C & the origin as its diameter.

I managed to find the radius which is √2 and the center which is (-2,3), I expressed C in cartesian form whihc i got (x+2)² + (y-3)²=2

Can anyone please help me in b & c. The answers together with how you ended up for that result are highly appreciated.

Thanks

part C: As far as giving you the answer and showing you how the answer was arrived will not happen on this forum. As the the name of the forum says we help you with your problem, not solve them for you. Here is a hint. You are given the endpoints of a diameter. The center is the midpoint of that diameter. Go from here.

Part B: What is the slope/derivative of y=-mx+c? Where is the derivative of (x+2)² + (y-3)²=2equal to the derivative of y=-mx+c

 

[asred9]

part C: As far as giving you the answer and showing you how the answer was arrived will not happen on this forum. As the the name of the forum says we help you with your problem, not solve them for you. Here is a hint. You are given the endpoints of a diameter. The center is the midpoint of that diameter. Go from here.

Part A: What is the slope/derivative of y=-mx+c? Where is the derivative of (x+2)² + (y-3)²=2equal to the derivative of y=-mx+c

Hi I cant seem to understand. Why we need the deriavite in part a ...

 

[Steven G]

Hi I cant seem to understand. Why we need the deriavite in part a ...

The straight line with equation y=-x+c is a tangent to the circle C
You want a line, of the form y=-x+b, to be tangent to the circle. That is why you want the derivative. After all, the derivative gives you the slope of the tangent line at a point. If this tangent line has any hopes of being y=-1mx+c then the derivative at this point on the circle better be equal to -1

 

[asred9]

The straight line with equation y=-x+c is a tangent to the circle C
You want a line, of the form y=-x+b, to be tangent to the circle. That is why you want the derivative. After all, the derivative gives you the slope of the tangent line at a point. If this tangent line has any hopes of being y=-1mx+c then the derivative at this point on the circle better be equal to -1

the derivative of y=-x+c =-1

 

[Steven G]

the derivative of y=-x+c =-1

Equal signs must be valid. The derivative of y does NOT equal -x+1 and -x+1 does not equal y. I will agree that the derivative of y = the derivative of -x+1 = 1.
So where does the derivative of the circle equal -1? Draw the line(s)! What is the equation(s) of the line(s)? What does c equal?

 

[asred9]

Equal signs must be valid. The derivative of y does NOT equal -x+1 and -x+1 does not equal y. I will agree that the derivative of y = the derivative of -x+1 = 1.
So where does the derivative of the circle equal -1? Draw this line! What is the equation of this line? What does c equal?

Hi,

I cat understand unfortunatly I lost around a week and a half about this topic since I was severely ill.

Can you pls shoe me an example .... even if this is not the answer to this.

Thanks since this is important for my schoel exams.

 

[Steven G]

Hi,

I cat understand unfortunatly I lost around a week and a half about this topic since I was severely ill.

Can you pls shoe me an example .... even if this is not the answer to this.

Thanks since this is important for my schoel exams.

Suppose you have the equation y=x^2 and you want to find the equation of the tangent line at x=2.

In order to find an equation of any line you either need two points OR a point and a slope. We will find the equation of this tangent line using one point and a slope.

A point has an x and y value. We know the x-value, we need to find the y-value. Since y=x^2 we know that the y-value is 2^2 or 4. So our point is (2,4).

We now need the slope of this line. Well y' = 2x and this IS OUR SLOPE FORMULA. You plug in an x-value into y' (or derivative formula) and out comes the slope of the tangent line. If x=2, then y'=2*2 =4. So the slope of the tangent line at x=2 is 4. So m=4.

The equation of the line which passes through (2,4) and whose slope is 4 is y=4x-4.


Another problem. What if we wanted to know the equation of the tangent line to y=x^2-5x and the slope of this line is to 3.

So again we need a slope and a point. Well we have the slope, it was given to be 3. Hence we need a point. This time we do not have the x-value of a point on the line. But we know that m=3 and this line is tangent to the curve. This means that we want y' to equal 3. y' = 2x-5 and it equals 3 when x=4. Then y=4^2-5*4=-4. So a point on the line is (4,-4)

The equation of this line is y=3x-16. Note that the slope is 3 and it crosses (4,-4) which is a point on the curve.

For part C. This is not a calculus problem at all. You are given the end points of a diameter of the circle. The center is in the middle of that diameter. The radius is the distance from the center to one of the endpoints of the diameter or half the distance of the diameter.
You need to know how to find the midpoint of a line segment given the endpoints and how to find the distance between two points.

Show us your work for additional help.