In the xy-plane, the point with coordinates (−6, −7) is the center of

Question

In the xy-plane, the point with coordinates (−6, −7) is the center of circle C. The point with coordinates (−6, 5) lies inside C, and the point with coordinates (8, −7) lies outside C. If m is the radius of C and m is an integer, what is the value of m ?

A. 12
B. 13
C. 14
D. 15
E. 16

A. 12
B. 13
C. 14
D. 15
E. 16

Archit3110 · Accepted Answer

Plot the coordinates as given :
we can say that point (-6,5) is at 12 distance from center which assures that radius >12
from point ( 8,-7) the distance is 14 and it is outside circle
so only integer value which can be the radius is 13 IMO B

somesh86 · Answer

can someone share the mathematical way of solving this question please?

philipssonicare · Answer

somesh86 
A radius is a line from the circle centre to the circumference. Create two lines: call the line from the circle centre to the first point 'Line A' and the line from the circle centre to the second point 'Line B'. The radius is between those two lengths.
A<m<B

Recognise that the first coördinate has the same x value as the circle centre. Therefore the line is merely the difference is y-values. 12
The second coördinate has the same y-value, but different x. Therefore the line is merely the difference in x-values. 14

12<m<14
It must be an integer. The only option is 13

B

Alternatively use the formula  \sqrt{(y2-y1)^2+(x2-x1)^2}  – we get y2, y1, x2, x1 just like when determining gradient in y=mx+b

Line A  \sqrt{(5--7)^2+(-6--6)^2} = \sqrt{(12)^2+(0)^2} = 12

Line B  \sqrt{(-7--7)^2+(8--6)^2} = \sqrt{(0)^2+(14)^2} = 14 
12 < m < 14.

gmataspirant2403 · Answer

Concept: Power of Point When substituting values of co-ordinates in equation of circle with center x1,y1: (x-x1^2) + (y - y1^2) = r^2 , if the value of LHSRHS (or r^2), the point is outside the circle. Solution: Make equation of circle ---> (y+7)^2 + (x+6)^2 = m^2 Put value of first point in LHS, and we know the point is inside the circle, hence, M>12. Similarly for second point, M<14. Only integral solution possible is M=13. Thanks

In the xy-plane, the point with coordinates (−6, −7) is the center of

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In the xy-plane, the point with coordinates (−6, −7) is the center of

Prep Toolkit

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