In the xy-coordinate plane, point P lies in the first quadrant such that the distance from… — GRE General Test

In the $xy$ -coordinate plane, point $P$ lies in the first quadrant such that the distance from the origin $O(0,0)$ to $P$ is $4\sqrt{3}$ , and line segment $OP$ forms a $30^\circ$ angle with the positive $x$ -axis. A circle centered at point $P$ with radius $10$ intersects the $y$ -axis at two distinct points. Which of the following statements are true? Select all such statements.

The $x$ -coordinate of point $P$ is $6$ .Answer
The length of the chord formed by the circle on the $y$ -axis is $16$ .Answer
C
The $x$ -coordinate of point $P$ is $2\sqrt{3}$ .
D
The length of the chord formed by the circle on the $y$ -axis is $4\sqrt{34}$ .
E
The perpendicular distance from point $P$ to the $y$ -axis is $2\sqrt{3}$ .

Answer

The correct statements are that the

x

-coordinate of point

P

6

and the length of the chord formed by the circle on the

y

-axis is

16

The horizontal leg of the

30^\circ-60^\circ-90^\circ

triangle gives the

x

-coordinate

x_P = 4\sqrt{3} \cdot \frac{\sqrt{3}}{2} = 6

. The perpendicular distance from center

P

to the

y

-axis is

6

. Using the Pythagorean theorem with circle radius

10

and distance

6

, half the chord length is

\sqrt{10^2 - 6^2} = 8

, yielding a full chord length of

16

Step-by-Step Solution

Determine the coordinates of point P using special right triangle ratio (30-60-90)

Point

P = (6, 2\sqrt{3})

Segment

OP

forms the hypotenuse of length

4\sqrt{3}

in a right triangle with the

x

-axis. The horizontal component is

x_P = 4\sqrt{3} \cos(30^\circ) = 4\sqrt{3} \cdot \frac{\sqrt{3}}{2} = 6

. The vertical component is

y_P = 4\sqrt{3} \sin(30^\circ) = 4\sqrt{3} \cdot \frac{1}{2} = 2\sqrt{3}

Find the perpendicular distance from P to the y-axis

Perpendicular distance

= 6

The

y

-axis is defined by the line equation

x = 0

. The perpendicular distance from

P(6, 2\sqrt{3})

x = 0

|6 - 0| = 6

Apply the Pythagorean theorem to calculate the chord length on the y-axis

Total chord length

= 16

A perpendicular segment from center

P

to the chord bisects the chord at midpoint

M(0, 2\sqrt{3})

. In right triangle

\triangle PMQ

formed by radius

PQ = 10

, distance

PM = 6

, and half-chord

MQ = d

d^2 + PM^2 = PQ^2 \implies d^2 + 6^2 = 10^2 \implies d^2 = 64 \implies d = 8

. The full chord length is

2d = 16

Key Concept

Properties of 30-60-90 special right triangles and application of the Pythagorean theorem to circle chord geometry

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