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Circle and point of tangency problem

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by alexandrabiorka » Fri Aug 01, 2014 8:03 am
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)
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by GMATinsight » Fri Aug 01, 2014 8:31 am
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)
Question : Co-ordinates of Point of Tangency ?

Given : Circle x^2 + y^2 = 25 and Line 3y = 4x + 25 are Tangent at only one point

For point of tangency the equation of line must satisfy the equation of circle
therefore,
x^2 + y^2 = 25 can be represented as
x^2 + [(4x + 25)/3]^2 = 25
i.e. 9x^2 + (4x + 25)^2 = 25*9
i.e. 9x^2 + 16x^2 + 625 + 200x = 225
i.e. 25x^2 + 200x + 400 = 0
i.e. x^2 + 8x + 16 = 0
i.e. (x+4)^2 = 0
i.e. x = -4
and Since 3y = 4x + 25
therefore, 3y = 4(-4) + 25
i.e. y = 3

Answer: Option A
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by GMATinsight » Fri Aug 01, 2014 8:41 am
ALTERNATE METHOD: (Perhaps the Best)

Start Checking Options
(The Correct Option will Satisfy the Equation of Circle as Well as Equation of Line as the point of Intersection falls on both the given curves)

Option 1) (-4,3)
For x^2 + y^2 = 25
(-4)^2 + (3)^2 = 16 + 9 = 25 SATISFIED

For 3y = 4x + 25
3(3) = 4(-4) + 25
9 = -16+25 SATISFIED BINGO!!!

Answer: Option A
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by Brent@GMATPrepNow » Fri Aug 01, 2014 9:25 am
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
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by GMATGuruNY » Fri Aug 01, 2014 10:33 am
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)
We can PLUG IN THE ANSWERS, which represent the point of tangency.
Since the point of tangency is where the line and the circle INTERSECT, the correct answer choice must lie on the line defined by 3y - 4x = 25.
Plugging the answer choices into 3y - 4x, we get:

A: (3*3) - (4 * -4) = 25.
B: (3*4) - (4 * -3) = 24.
C: (3 * 7/2) - (4 * -4) = non-integer.
D: (3*3) - (4 * -7/2) = 23.
E: (3*4) - (4 * -4) = 28.

Only A works.

The correct answer is A.
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