[Math] Finding the equation of a line whose segment is intercepted between axes

analytic geometrygeometry

The question is:
Find the equation of a line passing through $(-2, 5)$ and whose segment intercepted between axes in the 2nd quadrant is $7\sqrt{2}$

I have two graphs in mind but I don't know which one is correct.
The first graph is a line from point $(-2, 5)$ to the origin since the questions says "between axes". The second graph is a line from $(-2, 5)$ to the $x$ and $y$ axis. Which one is correct? I'm going to start with distance formula, right?

Best Answer

What we have is a line with positive slope, passing through the point $(-2, 5)$, with the length of the segment on this line from the point where it intersects the negative $x$-axis at $(x_0, 0)$ and the positive $y$-axis at $(0, y_0)$ equal to $7\sqrt 2$.

We know then, that $$x_0^2 + y_0^2 = (7\sqrt 2)^2 = 98\tag{1}$$

We also know that the distance from $(x_0, 0)$ to $(-2, 5)$ plus the distance from $(-2, 5)$ to the point $(0, y_0)$ is equal to $7\sqrt 2$: $$\sqrt {25 + (x_0+2)^2} + \sqrt{4+(y_0-5)^2} = 7\sqrt 2\tag{2}.$$

Solving $(1)$ and $(2)$ simultaneously gives us the real solutions $x_0 = -7$, and $y_0 = 7$. With these values, we must have that the slope of each of these "segments" comprising the entire length are equal: $$m = \frac{5-0}{-2 - x_0} = \frac{5-y_0}{-2-0}= 1$$

Now, given your point on the line $(-2, 5)$ and the slope of $m = 1$, we can construct the equation of the line: $$y - 5 = x + 2 \iff y = x+7$$

Related Solutions

[Math] Distance between a circle and a line segment

Here is an example Python code that solves the optimization problem suggested by @DeepSea

import casadi as ca
opti = ca.Opti()

# Line segment
t = opti.variable()
line_seg = opti.bounded(0, t, 1)
p0 = (-4.0, 0.5)
p1 = (-2.0, 1.5)
x = (1 - t) * p0 + t * p1

# Circle
y = opti.variable(2)
circ = y[0]**2 + y[1]**2 == 1

# Optimization problem
dist_squared = (x[0] - y[0])**2 + (x[1] - y[1])**2
opti.minimize(dist_squared)
opti.subject_to(line_seg)
opti.subject_to(circ)

# Solve
opti.solver('ipopt')
sol = opti.solve()

# Result
print(f"Distance = {sol.value(ca.sqrt(dist_squared))}")
print(f"Point on line segment: {sol.value(x)}")
print(f"Point on circle: {sol.value(y)}")

[Math] Can we take any decision about the quadrants, through which a line-segment passes, from its slope

I am assuming that you mean the y-cut is $0$. In other words, the equation of the line is of the form $$y=mx$$ where there is no constant term. I will also only consider lines and not line segments, as it is obvious that a line segment could be defined in such a way that it only falls in one quadrant (See Steven Stadnicki's comment). Then, I agree with your list of properties, except for the following amendments:

If $0<m<1$, the line lies at an angle of between $0 ^{\circ}$ and $45^{\circ}$ or between $180^{\circ}$ to $225^{\circ}$ to the horizontal.
If $m>1$, the line lies at an angle of between $45 ^{\circ}$ and $90^{\circ}$ or between $225^{\circ}$ and $270^{\circ}$ to the horizontal.
If $m<0$, the line lies at an angle of between $90 ^{\circ}$ and $180^{\circ}$ or between $270^{\circ}$ and $360^{\circ}$ to the horizontal.

Best Answer

Related Solutions

[Math] Distance between a circle and a line segment

[Math] Can we take any decision about the quadrants, through which a line-segment passes, from its slope

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