Hi Bunuel.
This question asks us if the circle intersects the y-axis, so how is the x-axis relevant?
Thanks!
THEORY
In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
(x-a)^2+(y-b)^2=r^2
![Image]()
This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.
If the circle is centered at the origin (0, 0), then the equation simplifies to: x^2+y^2=r^2
For more check: math-coordinate-geometry-87652.html
BACK TO THE ORIGINAL QUESTION
Does the curve (x - a)^2 + (y - b)^2 = 16 intersect the Y axis?
Curve of (x - a)^2 + (y - b)^2 = 16 is a circle centered at the point (a, \ b) and has a radius of \sqrt{16}=4. Now, if a, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis. So basically the question asks whether |a|>4: if it is, then the answer will be NO: the curve does not intersect with Y axis and if it's not, then the answer will be YES: the curve intersects with Y axis.
(1) a^2 + b^2 > 16 --> clearly insufficient as |a| may or may not be more than 4.
(2) a = |b| + 5 --> as the least value of absolute value (in our case |b|) is zero then the least value of a will be 5, so in any case |a|>4, which means that the circle does not intersect the Y axis. Sufficient.
Answer: B.
This question asks us if the circle intersects the y-axis, so how is the x-axis relevant?
Thanks!
Bunuel wrote:
catty2004 wrote:
I'm not sure the reasoning behind the red part in bold section......how did you get to 4b^2 - 4(1)(a^2 + b^2 -16) >=0 ?
THEORY
In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:
(x-a)^2+(y-b)^2=r^2
This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.
If the circle is centered at the origin (0, 0), then the equation simplifies to: x^2+y^2=r^2
For more check: math-coordinate-geometry-87652.html
BACK TO THE ORIGINAL QUESTION
Does the curve (x - a)^2 + (y - b)^2 = 16 intersect the Y axis?
Curve of (x - a)^2 + (y - b)^2 = 16 is a circle centered at the point (a, \ b) and has a radius of \sqrt{16}=4. Now, if a, the x-coordinate of the center, is more than 4 or less than -4 then the radius of the circle, which is 4, won't be enough for curve to intersect with Y axis. So basically the question asks whether |a|>4: if it is, then the answer will be NO: the curve does not intersect with Y axis and if it's not, then the answer will be YES: the curve intersects with Y axis.
(1) a^2 + b^2 > 16 --> clearly insufficient as |a| may or may not be more than 4.
(2) a = |b| + 5 --> as the least value of absolute value (in our case |b|) is zero then the least value of a will be 5, so in any case |a|>4, which means that the circle does not intersect the Y axis. Sufficient.
Answer: B.