show that the point (3/5, 4/5) is the closest point on the circle x^2 + y^2= 1 to the point on (3,4)
how do ido this
The closest point (x0,y0) on the circle to point (x1,y1) of the circle is if (x0,y0) lies on the radial line passing through (x1,y1).
To show this, first find the centre of the circle x^2+y^2=1 is a circle with radius 1 and centre (0,0).
The line through (0,0) and (3,4) is
L : y=(4/3)x (slope=(4-0)/(3-0)=4/3, and y-intercept=0)
Show that the point P(3/5,4/5) is on the line L as follows:
y=(4/3)x=(4/3)(3/5)=4/5 which equals the y-coordinate of the point P. So P lies on line L, thus (3/5,4/5) lies on the circle and is the closest point to (3/4)
you have a mystery number greater than 10 and less than 60 It has a remainder of 1 when divided by 6 and a remainder of 2 when divided by 5 what is the mystery number
To show that the point (3/5, 4/5) is the closest point on the circle x^2 + y^2 = 1 to the point (3, 4), we can use a direct method called the distance formula.
1. Begin by finding the equation of the line passing through the point (3/5, 4/5) and (3, 4). The equation can be found using the point-slope form: y - y1 = m(x - x1). Substituting the given points, we get:
y - 4/5 = (4/5 - 4)/(3/5 - 3) * (x - 3/5)
Simplifying this equation will give you the equation of the line.
2. Now, substitute the equation of the line back into the equation of the circle to find the points of intersection. Plug in the equation of the line for y in the equation of the circle:
x^2 + [(4/5 - 4)/(3/5 - 3) * (x - 3/5)]^2 = 1
Solve this equation to find the values of x, which will give you the x-coordinates of the points of intersection.
3. Once you have the x-coordinates of the points of intersection, substitute them back into the equation of the line to find the corresponding y-coordinates.
4. Calculate the distance between the points (3/5, 4/5) and the two points of intersection using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Compare the distances calculated. The point with the shorter distance is the closest point on the circle to the given point (3, 4). If the point (3/5, 4/5) has the shorter distance, it proves that it is the closest point.
By following these steps, you can demonstrate that the point (3/5, 4/5) is the closest point on the circle x^2 + y^2 = 1 to the point (3, 4).