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What is the radius equation of the circle x2 plus y2 -8x plus 4y equals 30 when touched by the tangent line y equals x plus 4?
x^2 + y^2 -8x + 4y = 30 & y = x + 4 Where the tangent touches The (x,y) values will be the same. So squaring the linear equation and substituting. y^2 = (x + 4)^2 Hence x^2 + (x + 4)^2 - 8x + 4(x + 4) = 30 Multiply out x^2 + x^2 + 8x + 16 -8x + 4 x + 16 = 30 Collect 'like terms'. 2x^2 + 4x + 32 = 30 2x^2 + 4x + 2 = 0 Divide by '2' x^2 + 2x + 1 = 0 Factor (x +1)^2 = 0 x = -1 When x = -1 y = 3 This is the point of contact of the tangent with the circumference. Since x = -8x , then the x-co-ord of the centre is '4' and y = 4y , y-co-ord of the centre is '-2'. In (x,y) tangent point is (-1,3) and circle centre is (4,-2). The radius is the distance between these two points. By Pythagoras. (-1 - 4)^2 + (3 - - 2)^2 = r^2 (-5)^2 + (5)^2 = r^2 25 + 25 = r^2 50 = r^2 r = sqrt(50) = sqrt(2 x 25) = 5sqrt(2) Done !!!! Hope that helps!!!
How do find x and y intercepts without a graph?
If for example: y = 2x+4 Then: y-2x = 4 And when the value of x is 0 then the y intercept is 4 And when the value of y is 0 then the x intercept is -2
When The circle below is centered at the point (2 -3) and has a radius of length 4. What is its equation?
Using Pythagoras. ( x - 2)^" + ( y - - 3)^2 = 4^2 Becomes (x - 2)^2 + (y + 3)^2 = 4^2 This can be expanded to : - # x^2 - 4x + 4 + y^2 + 6y + 9 = 16 Collecting terms. x^2 + y^2 - 4x + 6y - 3 = 0
How do you find the width and length of a rectangle knowing ONLY the diagonal and the area?
For e.g we draw a rectangle with length and width of 4cm and 3cm respectively. So the diagonal would be of 5cm(using Pythagoras theorem) and area 12cm^2. Now using algebra we can find the dimensions of the rectangle,Look below: let length=x let width=y so: x*y=12----(1) x^2+y^2=5^2----(2) x^2+y^2=25 Now simplifying the above equations: x^2=25-y^2 ---->x^2*y^2=144 ---->(25-y^2)*y^2=144 25y^2-y^4=144 y^4-25y^2+144=0 let y^4=z^2 so: z^2-25z+144=0 by simplifying the above quadratic equation: z=(25+(25^2-4(1)(144))^(1/2))/2(1) , z= (25-(25^2-4(144))^(1/2))/2 z=16, z=9 when z=16, when z=9 so y=4 ,so y=3 Now verifying which magnitude is correct for y: 12/4=x ,12/3=x x=3 not correct, because y>x x=4 is correct, because x>y
What is the equation of a circle with center (1 -4) and radius 2?
Using (x-h)^2+(y-h)=r^2 sub in -1 and 2 for h and k get sqrt of 4 for r anser is therefore (x+1)^2+(y-2)=2^2 Have and Wonderful Day, Danny
