Dizzy Planets Ellie is a keen astronomer. One day, whilst stargazing, she spots Mercury and Venus.
Being a keen astronomer, she knows that Venus
takes 225 Earth days to go around the sun and
Mercury takes 88 Earth days for the same journey.
Assuming circular orbits, and that Venus and
Mercury are in alignment (relative to the sun) at
00:00 on 1st January 2021, how many times will
they be aligned (V, M, S in that order) during that
Earth year?
On which days will these alignments occur?

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12

The answer would be -3.9> -4.5