You are given an ellipse and a point P on the ellipse. Find $F _ { 1 } P$ and $F _ { 2 } P$ , the lengths of the focal radii. $\frac { x ^ { 2 } } { 87 ^ { 2 } } + \frac { y ^ { 2 } } { 29 ^ { 2 } } = 1 ; P ( 63,20 )$
A) $F _ { 1 } P = 87 + 42 \sqrt { 2 } , F _ { 2 } P = 87 - 42 \sqrt { 2 }$
B) $F _ { 1 } P = 841 + 42 \sqrt { 2 } , F _ { 2 } P = 29 + 42 \sqrt { 2 }$
C) $F _ { 1 } P = 29 + 43 \sqrt { 2 } , F _ { 2 } P = 841 - 43 \sqrt { 2 }$
D) $F _ { 1 } P = - 87 - 42 \sqrt { 2 } , F _ { 2 } P = 87 + 42 \sqrt { 3 }$
E) $F _ { 1 } P = - 29 + 43 \sqrt { 2 } , F _ { 2 } P = - 841 - 43 \sqrt { 2 }$

Question

Quizplus · Accepted Answer

The ellipse equation is given by $\frac{x^2}{87^2} + \frac{y^2}{29^2} = 1$, which means the semi-major axis $a = 87$ and the semi-minor axis $b = 29$. The focal distance $c$ can be found using $c^2 = a^2 - b^2$, which gives $c = \sqrt{87^2 - 29^2} = \sqrt{7569 - 841} = \sqrt{6728} = 82$. The foci are at $(\pm c, 0)$ or $(\pm 82, 0)$. The distance from a point $P(x, y)$ on the ellipse to a focus is given by the distance formula. For $P(63, 20)$, the distances to the foci are $F_1P = \sqrt{(63 + 82)^2 + 20^2} = \sqrt{145^2 + 20^2}$ and $F_2P = \sqrt{(63 - 82)^2 + 20^2} = \sqrt{(-19)^2 + 20^2}$. Simplifying these expressions directly to match the given options is complex without exact arithmetic, but the principle involves recognizing that the sum of distances from any point on the ellipse to the foci is constant and equals $2a$, which is $2 \times 87 = 174$. The correct answer must reflect this relationship, and among the options, only A correctly implies a relationship that could be consistent with the properties of an ellipse, despite the direct calculation not being shown. The other options either do not match the expected format of an answer or contain arithmetic errors.

You Are Given an Ellipse and a Point P on the Ellipse