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Circle ABCD in the diagram above is defined by the equation x2+y2=25.

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Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   24 Oct 2015, 06:11
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Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. 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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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   24 Oct 2015, 08:30
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shasadou wrote:
Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. 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)


Property of 2 mutually perpendicular lines with slopes m1 and m2 ---> m1*m2=-1

Also, the radius drawn from the center of a circle to the point of tangency, is perpendicular to the tangent at the point of tangency.

Thus, equation of line perpendicular to 3y=4x+25 (slope = 4/3)---> y = mx+c with m*(4/3)=-1---> m = -3/4 and it passes through (0,0) giving you c=0 .

Thus the equation of line perpendicular to the given line ---> y = -3/4 * x . Now solve this equation with 3y=4x+25 to get the point of tangency as (-4,3)

Thus, A is the correct answer.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   28 Jan 2016, 01:21
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The point of tangency would satisfy both the equations.

Only (–4, 3) does that. Beware of choice B :P

Option A is the correct choice!
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   24 Oct 2015, 07:48
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We are given the equation of a line: 3y = 4x + 25 => y = (4/3)x + 25/3
Substituting the value of y in the equation of circle, we get:
x^2 + [ (4/3)x + 25/3 ]^2 = 25
=> x^2 + (16/9)x^2 + (200/9)x + 625/9 = 25
Multiplying by 9 on both sides,

9x^2 + 16x^2 + 200x + 625 = 225
25x^2 + 200x + 400 = 0
x^2 + 8x + 16 = 0

This gives us the value of x = -4
Substituting this value in the equation of line we get,
3y = 4*(-4) + 25 = -16 + 25 = -9
So, y = -3
And out point is (x,y) = (-4,-3)

Ans: A.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   28 Jan 2016, 02:18
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Point should satisfy both the equations i.e, for circle and the line. Plug in the values to get the answers. Its A.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   16 Apr 2016, 09:06
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Great explanations all there by intelligent folks. I just looked up the coordinates :) That also serves some hints
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   16 Jun 2017, 02:02
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shasadou wrote:

Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. 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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Attachment:
Geometry_Img69.png


There's a couple of formulas we should have at our disposal in order to solve this question; remember that the slope of the tangent line is the negative reciprocal of the slope of the radius of the circle. Well, we know the center of this circle (0,0). Hence

(x-0)^2 + (y-0)^2 = 25

The two values of x and y must satisfy this equation and also the equation 3y= 4x + 25

Thus

"A"
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   01 Jul 2017, 00:05
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Circle ABCD in the diagram above is defined by the equation x^2+y^2=25. 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)

A point if tangent would satisfy both the equations of line and circle

\(x^2+y^2=25\) and \(3y=4x+25\)

The only point from the options that satisfy the equation is A \((–4, 3)\)

Hence, Answer is A
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   01 Jul 2017, 00:25
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as the point of tangency is a common point, it must satisfy both the equations.

only option A and B satisfy the circle equation n this means these two points lie on the circumference of the circle.

now we can get the right answer by putting the values of the x and y co-ordinates on the tangent equation.

only option A satisfies this equation.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   22 Apr 2020, 14:12
The options should satisfy the Equation of Circle thus eliminating c and d options.
The option a and b should satisfy equation of line thus eliminating option b.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   26 Aug 2021, 02:26
You can use this shortcut, from the equation of the circle it is certain that the radius is 5. Now the distance between the origin and the tangent is 5. Options A and B only satisfies this condition so eliminate all other options. Now between A and B, Option A satisfies the tangent line equation which means, point of intersection of tangent and circle is option A.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   30 Sep 2021, 09:11
We can solve the question by using distance formula as well.

Distance from the tangent point to the centre of the circle has to be 5 =radius.

Hence root { (-4_0)sq + (3-0)sq} = 5
Satisfied hence ans is A.
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink] New post   18 Apr 2024, 19:33
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Re: Circle ABCD in the diagram above is defined by the equation x2+y2=25. [#permalink]
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