Find Slope Intercept

Question

(a) Straight line passes through the point (-2,8) and is parallel to the line x+2y=4

Slope Intercept Equation is:

(b) Straight line passes through the point (-1,9) and is perpendicular to the line 5x+y=4

Slope Intercept Equation is:

(c) Determine the center and the radius of the circle x^2+y^2-2x+5y+5=0

The Center = ___ Radius = ___

Answer 1

The slope intercept form of a line is y = mx + b.  The equation x + 2y=4 is changed to the slope intercept form by solving the equation for y.  So 2y = -x + 4.  You need to divide the right side of the equation by 2 to solve for y, so the answer is y = -1/2x + 2 (m= -1/2, b=2).

 

To solve b, the equation (5x+y=4) needs to be slope intercept form (y=mx+b), in order to find the slope.  To do this, you need to solve for y.  So y = -5x + 4.  The equation of perpendicular line will have the negative inverse of the slope (m) from first equations, so the new equation will be y = 1/5x + b since the negative reciprocal of -5 is 1/5.  You can find b by substituting the x and y from the point (-1,9) and then solve for b.  So 9 = 1/5(-1) + b (since x =-1 and y=9).  The b = 9 1/5 or 46/5.  So the new equation is y = 1/5x + 46/5.

 

The formula for a circle is (x-h)^2 + (y-k)^2 = r^2.  So the given equation, X^2+y^2 -2x+5y = 0, needs to be arranged in the circle formula format by completing the squares.  So the first step is to combine the x's and y's on one side of the equation, x^2-2x+y^2+5y = -5.  The you can complete the squares by adding the necessary values to both sides of the equation.  ( x^2 + 2x + 1) + (y^2 +5y + 6.25) = 2.25 ( -5 + 1 + 6.25 ).  To complete the square the value of the middle term is divided by half and then square, so 1 square is one and 2.5 squared is 6.25)  So the finished equation is (x-1)^2 + (y+2.5)^2 = 2.25  The center of the circle is h,k which is (1, -2.5) and the radius is 1.5 (the square root of 6.25).