# Tutor profile: Pierre E.

## Questions

### Subject: SAT II Mathematics Level 2

A circle has parametric equations $$ x= 5cos(t)$$ and $$y = 5 sin(t) $$ , where t is the parameter. Find the diameter of this circle

1) Recall $$ (x-h)^2 + (y-k)^2 = R^2 $$ is the equation of the circle 2) Find $$ x^2 =[5cos(t)]^2 =25cos^2(t) $$ 3) Find $$ y^2 =[5sin(t)]^2 =25sin^2(t) $$ 4) therefore, $$ x^2 +y^2 = 25cos^2(t)+ 25sin^2(t) = 25 [cos^2(t)+ 25sin^2(t)] $$ 5) Using the trig identity $$ cos^2(t)+ sin^2(t) =1 $$, we get $$ x^2+y^2 =25 $$ 6) Since $$ R^2=25 $$ , $$ R = \sqrt25= 5 $$ 7) Finally Diameter = 2 radius, hence diameter = 10

### Subject: SAT

The diameter of a circle graphed in the xy-plane has endpoints at (-23,15) and (1,-55). Which of the following is an equation of the circle?

1) The formula for a circle with a center at (h,k) and a radius R is $$ \ (x-h)^2 +(y−k)^2 =R^2$$ 2) Let's begin by finding the center. The center will be at the midpoint of the diameter. (I) The h-coordinate will be halfway between the two x-values of the diameter. $$ \frac{-23+1}{2} =-11$$ (II) The k-coordinate will be halfway between the two y-values of the diameter. $$ \frac{-55+15}{2} =-20$$ 3) The radius is the distance from the center to any endpoint. So, using the distance formula: $$ R^2=(x-h)^2+(y-k)^2 = (-23--11)^2 +(15--20)^2 = (-12)^2 +(35)^2 = 1369 $$ . 4) Let's put the center and the radius into our equation for a circle $$ (x+11)^2+(y+20)^2 =1369 $$

### Subject: ACT

If the mean of the numbers {4, 16, 10, x, 8, 20} is 12, what is the difference between the mean and the median?

1) write the equation of the mean: (4+16+10+8+20+x)/6=12 or (58+x)/6=12 2) Solve for x: 58 + x =6*12 or 58 + x = 72 hence 58-58 + x = 72-58 hence x = 14 3) write the six numbers in ascending order: 4, 8, 10, 14, 16, 20 4) Find the median as (10 + 14)/2 =12 5) Compute the difference between the mean and the median: 12-12 = 0