Problem Solving

alexandrabiorka wrote:Circle ABCD is defined by the equation x^2 + y^2 = 25 on a coordinate plane. Line segment EF is defined by the equation 3y = 4x + 25 and is tangent to circle ABCD at exactly one point. What is the point of tangency?

(A) (-4,3)
(B) (-3, 4)
(C) (-4, 7/2)
(D) (-7/2, 3)
(E) (-4, 4)

The point of tangency will be a point that is on the circle defined by the equation x² + y² = 25 AND on the line defined by the equation 3y = 4x + 25.
So, that coordinates of that point will SATISFY both equations: x² + y² = 25 AND 3y = 4x + 25

Let's deal with x² + y² = 25 and SCAN the answer choices
A) (-4)² + 3² = 25 GOOD
B) (-3)² + 4² = 25 GOOD
C) Notice that (-4)² is an integer and (7/2)² is NOT an integer, so there's no way that (-4)² + (7/2)² can equal 25 (which is an integer). So, it's the coordinates (-4, 7/2) CANNOT satisfy the equation x² + y² = 25
D) using the same logic as above, the coordinates (-7/2, 3) CANNOT satisfy the equation x² + y² = 25
E) (-4)² + 4² = 32 NO GOOD

So, we know the answer is EITHER A or B

IMPORTANT: At this point, we'll test EITHER A or B to see if it satisfies the equation 3y = 4x + 25
We'll test only one of these answer choices. So, if we test A and it works, then A is the correct answer. If we test A and it doesn't work, then we'll AUTOMATICALLY select B, since it MUST be the answer (now that we've eliminated C, D and E)

Test answer choice A to see if it satisfies the equation 3y = 4x + 25
3(3) = 4(-4) + 25 Perfect!

The correct answer is A

Cheers,
Brent