seekmba wrote:
How many points of intersection does the curve x^2 + y^2 = 4 have with line x+y =2 ?
A. 0
B. 1
C. 2
D. 3
E. 4
did not understand the explanation.
Curve \(x^2 + y^2 = 4\) is a circle with radius 2 and the center at the origin. A line cannot have more than 2 points of intersection with a circle, so we can eliminate choices D and E. To answer the question, we can either draw a sketch or solve the system of equations. The second approach gives\((2-y)^2 + y^2 = 4\)or \(2y^2 - 4y + 4 = 4\) or \(y^2 - 2y = 0\) from where y=0 and y=2. Thus, the line and the circle intersect at points (2,0) and (0,2).
The curve x^2 + y^2 = 4 is a circle with center at (0, 0) and with a radius of 2. Since a line and a circle can intersect at 0, 1, or 2 points, we immediately eliminate D and E.
The best approach to solving this question is to draw the given circle and the line. When we do that, we will notice that both intercepts of x + y = 2 (which are (0, 2) and (2, 0)) are contained in the given circle, which means the line and the circle intersect at two points.
Though time consuming, it is also possible solve this question using algebra. Let's write y = 2 - x and substitute in the equation x^2 + y^2 = 4:
\(\Rightarrow\) x^2 + y^2 = 4
\(\Rightarrow\) x^2 + (2 - x)^2 = 4
\(\Rightarrow\) x^2 + 4 - 4x + x^2 = 4
\(\Rightarrow\) 2x^2 - 4x = 0
\(\Rightarrow\) x^2 - 2x = 0
\(\Rightarrow\) x = 2 or x = 0
Substituting x = 2 in x + y = 2 yields y = 0, so we get the point (2, 0). Similarly, substituting x = 0 in x + y = 2 yields y = 2, so we get the point (0, 2). So the line and the circle intersect in two points.
Answer: C