In the xy plane, a circle is drawn with the center at (3, -2). The cir

how did you get ao and ob

Bunuel IanStewart even i am confused regarding the question, can you explain this with a better POE.

summerbummer


Perpendicular from the center to the chord bisects the chord.

Hence, AX = XB = √14

OX = (3-0) = 3

AO = AX - OX = √14 - 3

OB = XB + OX = √14 + 3

If you still have any doubt, you can ask.

Is there anyone to share easiest way, please? I really appreciate your help.

Hi,

I got the correct answer by using some logic of my own. I'd like to share the same.

The question says distance between the "Points" of intersection and Y Axis.

Please use the image shared by nick1816 as a reference towards my explanation.

The center has to be (3,-2) so imagine a center at (3,-2).
Now try to draw a circle from that point. Remember the circle intersects at 2 points on the X Axis which are 2√14 apart. But that can be anywhere. So assume any radius for this circle.

For example 6 units.
So go 6 units right from 3 on the X Axis. [3 is the original center so imagine using a geometry compass from (3,-2) and take 6 units and draw a circle. You'll intersect X Axis at 9]

This is one point of intersection.

Now go the same 6 units left from 3 on the X axis. You'll get -3 on the negative side of X Axis.

This is second point of intersection.


Now the distance between these two and the Y Axis.

Distance from 9 (Positive X Axis) to Y Axis = 9
Distance from Y Axis to -3 (Negative X Axis) = -3

9 + (-3) = 6

Take any number as radius and do the above calculation, you'll always get 6 as the answer.


This is purely based on my logic. I might be wrong.


My heart says this is correct but my brain says something is wrong or missing.

Therefore, I humbly request Bunuel, chetan2u, nick1816, yashikaaggarwal to check my explanation and let me know how close am I to the correct logic.

It will be great if any of you or any of your expert friends gave us some insights.

DON'T RELY ON THIS UNLESS ANY ONE OF THE EXPERTS CONFIRM IT.

Please don't mind the shape of my circle in the picture that I've attached.

Lastly, I'm sorry if my logic has hurt any math enthusiast or math lover due to its potential loopholes.


Thank you.

Posted from my mobile device

As IanStewart thankfully pointed out, the question is asking for the distance between the Y-Intercepts (at least that is what the credited response of 6 gives us).


Because a circle is a collection of equidistant points from the center, the circle will come up and intersect one point on the X Axis (point A) and come around counterclockwise and intersect another point on the X Axis (point B) —— such that every point on the circumference is equidistant from the center.

Thus, Point A and Point B will each be a distance of R = radius from the center of the Circle, point O at (3 , -2)


Rather than use the circle, we can use the Rules and Symmetry of an Isosceles triangle when an altitude is dropped from the apex vertex.


Connect 2 radii from the Center O to points A and B.


The distance between the two X intercepts is 2 * sqrt(14) and this will be the Base of the triangle = Side AB

OA = OB = radius = R ———> are the other 2 equal sides of the Isosceles Triangle


Rule: the height originating from the apex vertex O at (3 , -2) and drawn perpendicular to base side AB, as a line of symmetry, will also be the Median of the isosceles triangle and Bisect Side AB.

Call the point at which this height bisects side AB —- point D

Since the distance between the 2 X intercepts (points A and B) is given as 2 * sqrt(14)

AD = sqrt(14) = DB

The length from center (3 , -2) to point D on the X Axis is given by the vertical distance from the line Y = 0 to the line Y = -2 ————> 2 units

We can then use the Pythagorean Theorem to find the radius of the circle (which is the two equal sides, OA and OB, of the Isosceles triangle)

(AD)^2 + (2)^2 = (R)^2

(Sqrt(14))^2 + (2)^2 = (R)^2

18 = (R)^2


Lastly, we can set up the equation for a Circle and find the two Y intercepts.

The equation of this circle, with center at point O (3 , -2), is given by:

(x - 3)^2 + (y + 2)^2 = 18


The Y intercepts will occur at the point where the X coordinate is 0:

(0 - 3)^2 + (y + 2)^2 = 18

9 + (y)^2 + 4y + 4 = 18

(y)^2 + 4y - 5 = 0

(y + 5) (y - 1) = 0


The Y intercepts - the points at which the circle will intersect the Y Axis - will occur at:

(0 , -5)

And

(0 , +1)

This is a vertical distance of 6 units

Answer: 6



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