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SAT - Math - Coordinate geometry

A line in the xy-plane passes through the origin and has a slope of 1/7 Which of the following points lies on the line? (more)

A line in the xy-plane passes through the origin and has a slope of 1/7 Which of the following points lies on the line?

A) (0, 7)
B) (1, 7)
C) (7, 7)
D) (14, 2)
E) (7, 14)

Given

slope (m) = 1/7

intercept (c) = 0 (as the line passes throught the origin)

Equation of line: y = mx + c

y = 1/7 x 

7y = x

only point that satisfies the equation is (14, 2)

The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle? (more)

x2 20x y2 16y = −20

The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?
A) (−20,−16)
B)(−10, −8)
C) (10, 8)
D) (20,16)

x2 + 20x + y2 + 16y = −20

Writing the equation in standard form

x+ 20x + y2 + 16y + 20 = 0

for a circle x2 + y2 + 2gx + 2hy + d = 0 center is (-g, -h)

center of the circle would be: (-10, -8)

Line l passes through the origin and makes 30 degrees with the positive x axis. (more)

Line l passes through the origin and makes 300 with the positive x axis.

find the equation of line l.

a) y = x\2
b) y = x\root 2
c) y= x\root 3
d)y= (root 2) x
e) y= (root 3) x

general equation of a line is y = mx + c where m is gradient and c is y intercept m = tan(30) m = 1/√3 coordinates if the origin is (0,0) Putting that in the gen eqn We have 0 = (1/√3) . 0 + c C = 0 So then , the eqn of the line is y = (1/√3)x

3 points lie on an xy plane (more)

3 points lie on an xy plane, with each point representing where a particular person lives. Amy lives at point (2,3); Brian lives at point (0,7); and Claire lives at point (-2,3). They all wish to meet up with each other for lunch, but at a location that is equidistant from each of their respective houses. At what point do they meet?
a. (0,4.5)
b. (0,13/3)
c. (0,3)
d. (1,5)
e. (-1,5)

 

The point equidistant from 3 points in a coordinate plane would be the circumcenter of the triangle formed by the three points.

Circumcenter is the intersection point of the perpendicular bisectors of the sides of the triangle.

Equation perpendicular bisector of (2,3) and (0,7)

(x-2)2 + (y-3)2 = (x-0)2 + (y-7)2

simplyfing and canceling second order terms

4 - 4x + 9 - 6y = 49 - 14y

4x - 8y +36 = 0

Similarly Equation of perpendicular bisector of  (2,3) and (-2,3)

(x-2)2 + (y-3)2 = (x+2)2 + (y-3)2

(x-2)2 = (x+2)2

only solution x = 0

simultaneously solving both equations

put x = 0 in first equation

8y = 36

y = 9/2

required point: (0,4.5)