Tangents to an Ellipse

A definition for an ellipse is:
Given a circle $c$ with centre $M$ and radius $r$ and a point $G$ such that $d(G,M) \lt r$, the locus of the points that are equidistant from $c$ and $G$ form an ellipse.

The construction of the points of the ellipse is shown below.

Given are the points $M(-2000,1500)$ and $G(8000,1500)$.
Given is also the circle $c$ with centre $M$ and radius $15000$.
The locus of the points that are equidistant from $G$ and $c$ form an ellipse $e$.
From a point $P$ outside $e$ the two tangents $t_1$ and $t_2$ to the ellipse are drawn.
Let the points where $t_1$ and $t_2$ touch the ellipse be $R$ and $S$.

For how many lattice points $P$ is angle $RPS$ greater than $45$ degrees?

To solve this problem, we need to find the equations of the tangents $t_1$ and $t_2$ to the ellipse and then check the angle $RPS$ for each lattice point $P$.

First, we can start by finding the equation of the ellipse using the given center and radius. In general, the equation of an ellipse centered at $(h, k)$ with semi-major axis $a$ and semi-minor axis $b$ can be written as:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

For our specific ellipse, the center is $M(-2000,1500)$ and the radius is $15000$. Since the radius is the length of the semi-major axis, $a = 15000$. The length of the semi-minor axis can be calculated using the distance formula as $b = \sqrt{(G_x – M_x)^2 + (G_y – M_y)^2}$, where $G(8000,1500)$ and $M(-2000,1500)$. Therefore, $b = 10000$.

Substituting the values into the equation, we have:

$\frac{(x+2000)^2}{15000^2} + \frac{(y-1500)^2}{10000^2} = 1$

Now, we need to find the equations of the tangents $t_1$ and $t_2$ to the ellipse. The general equation of a tangent to the ellipse at point $(x_0, y_0)$ is given by:

$\frac{(x-x_0)}{a^2} + \frac{(y-y_0)}{b^2} = 1$

To find the points of tangency $R$ and $S$, we need to find the $x$-coordinates of the intersection between $t_1$ and $t_2$ and the ellipse.

The slope of the tangent line at a point $(x_0, y_0)$ on the ellipse can be calculated using the formula:

$m = -\frac{b^2(x_0-h)}{a^2(y_0-k)}$

Now let’s proceed to implement the code in Python to find the number of lattice points $P$ where angle $RPS$ is greater than $45$ degrees.

“`python
import math

# Given points and ellipse parameters
M = (-2000, 1500)
G = (8000, 1500)
a = 15000
b = math.sqrt((G[0] – M[0])**2 + (G[1] – M[1])**2)

# Equation of the ellipse
def ellipse_equation(x, y):
return ((x + 2000)**2)/(a**2) + ((y – 1500)**2)/(b**2)

# Function to calculate slope of the tangent line at a point
def tangent_slope(x, y):
return -b**2*(x + 2000)/(a**2*(y – 1500))

# Function to calculate angle RPS in degrees
def angle_RPS(P, R, S):
PR = math.sqrt((P[0] – R[0])**2 + (P[1] – R[1])**2)
PS = math.sqrt((P[0] – S[0])**2 + (P[1] – S[1])**2)
RS = math.sqrt((R[0] – S[0])**2 + (R[1] – S[1])**2)
return math.degrees(math.acos((PR**2 + PS**2 – RS**2) / (2 * PR * PS)))

# Count of lattice points where angle RPS > 45 degrees
count = 0

# Loop through all lattice points
for x in range(-20000, 20001):
for y in range(-20000, 20001):
# Check if point P is outside the ellipse
if ellipse_equation(x, y) > 1:
# Calculate slope of tangent line at point P
m = tangent_slope(x, y)
# Calculate discriminant to determine number of intersection points
discriminant = b**2 – (a**2 * m**2)
# Check if the discriminant is positive (tangents intersecting the ellipse)
if discriminant > 0:
# Calculate x-coordinates of intersection points (tangents with ellipse)
x1 = (-a**2 * m * b + math.sqrt(discriminant)) / (a**2 * m**2 + b**2)
x2 = (-a**2 * m * b – math.sqrt(discriminant)) / (a**2 * m**2 + b**2)
# Check if the intersection points are valid (tangents touching the ellipse)
if abs(ellipse_equation(x1, m*x1+y)) < 1e-6 and abs(ellipse_equation(x2, m*x2+y)) < 1e-6: # Calculate angle RPS and check if it is greater than 45 degrees if angle_RPS((x, y), (x1, m*x1+y), (x2, m*x2+y)) > 45:
count += 1

print(“Number of lattice points where angle RPS > 45 degrees:”, count)
“`

This code will iterate through all the lattice points from (-20000, -20000) to (20000, 20000) and check the conditions to count the number of points where the angle $RPS$ is greater than $45$ degrees.

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