a line is drawn from the origin to a point on the ellipse 9x^2+25y^2=225.find the equation of the locus of midpoint of the line.
a point (x,y) on the ellipse is
(x,(1/5)√(225-9x^2))
halfway there is
(x/2,(1/10)√(225-9x^2))
x^2/4 + y^2/b^2 = 1
x^2/4 + (225-9x^2)/100b^2 = 1
25b^2x^2/4 + 225-9x^2 = 100b^2
(25b^2/4 - 9)x^2 = 100b^2-225
25b^2/4 = =9
b^2 = 36/25
x^2/4 + (25/36)y^2 = 1
9x^2 + 25y^2 = 36
Hmmm. Not what I expected. 36 seems too small. Better check the distance of the two points on some radius.
To find the equation of the locus of the midpoint of a line that is drawn from the origin to a point on the ellipse 9x^2 + 25y^2 = 225, we can solve this problem step-by-step.
Step 1: Write the equation of the line between the origin (0,0) and a point on the ellipse in slope-intercept form (y = mx + b).
Let the point on the ellipse be (x, y). The slope of the line passing through the origin and (x, y) can be found as follows:
slope (m) = (y - 0) / (x - 0) = y / x
The line's equation is:
y = (y / x) * x + b
Simplifying this, we have:
y = y + b
This implies that b = 0.
So, the equation of the line is:
y = y / x * x
Step 2: Find the coordinates of the midpoint on the line.
The midpoint coordinates are obtained by averaging the x and y coordinates of the endpoints (0,0) and (x, y):
(x_m, y_m) = ((0 + x) / 2, (0 + y) / 2)
= (x/2, y/2)
Step 3: Substitute the midpoint coordinates (x/2, y/2) into the equation of the line to find the equation in terms of the midpoint.
Substituting (x/2, y/2) into the equation y = y / x * x, we get:
y/2 = (y/2) / (x/2) * (x/2)
y/2 = y/x * (x/2)
Simplifying this further, we have:
y/x = 1/2
Step 4: Convert the midpoint equation into standard form.
To convert the midpoint equation into standard form (Ax + By + C = 0), we multiply both sides by 2x:
y = (x/2)
2xy = x
2xy - x = 0
x(2y - 1) = 0
This implies that either x = 0 or 2y - 1 = 0.
If x = 0:
0(2y - 1) = 0
0 = 0
This shows that x = 0 is a valid solution.
If 2y - 1 = 0:
2y = 1
y = 1/2
This shows that y = 1/2 is a valid solution.
So, the equation of the locus of the midpoint is:
x = 0 or y = 1/2
To find the equation of the locus of the midpoint of the line drawn from the origin to a point on the given ellipse, we can follow these steps:
Step 1: Find the equation of the line passing through the origin and a point on the ellipse.
Step 2: Determine the coordinates of the midpoint of the line.
Step 3: Express the coordinates of the midpoint in terms of the parameter used to describe the point on the ellipse.
Step 4: Eliminate the parameter to find the equation of the locus.
Let's go through each step in detail:
Step 1: Find the equation of the line passing through the origin and a point on the ellipse.
The equation of a line passing through the origin and a point (x, y) can be written in the form:
y = mx
To find the slope (m), we can differentiate the equation of the ellipse with respect to x. So, differentiate both sides of the given equation:
18x + 50yy' = 0
Solving for y', we get:
y' = -18x / (50y)
Since the slope of the line is given by dy/dx, and y = mx, we can equate dy/dx to m to find the slope of the line:
m = -18x / (50y)
Now, substitute the coordinates of the point (x, y) on the ellipse into this equation to find the slope of the line passing through the origin and that point.
Step 2: Determine the coordinates of the midpoint of the line.
The midpoint of a line segment is given by the average of the x-coordinates and the average of the y-coordinates. Since the origin (0, 0) is one endpoint of the line, the midpoint's x-coordinate will be half of the x-coordinate of the point on the ellipse, and the y-coordinate will be half of the y-coordinate.
So, the coordinates of the midpoint can be written as:
(x/2, y/2)
Step 3: Express the coordinates of the midpoint in terms of the parameter used to describe the point on the ellipse.
Let's assume the parameter used to describe the point on the ellipse is 't'. We can express x and y in terms of 't' using the parametric equations for the ellipse:
x = a * cos(t)
y = b * sin(t)
For the given ellipse, since the coefficients of x^2 and y^2 are not equal, we can determine that a = 5 and b = 3. Therefore, the parametric equations become:
x = 5 * cos(t)
y = 3 * sin(t)
Step 4: Eliminate the parameter to find the equation of the locus.
Now, substitute the expressions for x and y from Step 3 into (x/2, y/2) to get the coordinates of the midpoint in terms of 't':
(x/2, y/2) = (5 * cos(t) / 2, 3 * sin(t) / 2)
Finally, eliminating 't', we can write the locus equation by squaring both sides of the x-coordinate expression and the y-coordinate expression separately:
x^2 / 4 = (5 * cos(t))^2 / 4
y^2 / 4 = (3 * sin(t))^2 / 4
Simplifying these equations:
x^2 = 25 * cos^2(t)
y^2 = 9 * sin^2(t)
Therefore, the equation of the locus of the midpoint of the line is:
25 * cos^2(t) + 9 * sin^2(t) = 225
This is the equation of the locus of the midpoint of the line.