conic sections
question 1:
verify that the point (3,2) lies on the circle x^2+y^2-8x+2y+7=0, and find the equation of the tangent at this point.
question 2:
find the length of the tangents from the point (5,7) to the circle x^2+y^2-4x-6y+9=0
question 3:
find the equation of the common chord of the circles x^2+y^2-6x+2y+1=0 and x^2+y^2+4x-2y-13=0
#1. Have you tried plugging x=3 and y=2 into the equation? Do it and verify that the equation is satisfied there.
Then get the circle's slope dy/dx at (3,2) by implicit differentiation.
Once you know one point on the line (3,2), and the slope at that point, the equation of the line can easily be written.
(y-2) = (x-3)*dy/dx
#2 Rewrite the circle equation
(x-2)^2 -4 + (y-3)^2 -9 + 9 = 0
(x-2)^2 + (y-3)^2 = 4
That tells you the circle's center location and radius. Plot it and the point (5,7) on a graph. Draw a line from (5,7) to the circle's center at (2,3). From the length of that line (a hypotenuse) and the radius of the circle, use the Pythagorean theorem to get the length of the tangent. A tangents, radius to the tangency point, and hypotenuse form a right triangle.
3. That's all I have time for. Please show your own work on future posts
Question 1:
To verify that the point (3,2) lies on the circle x^2 + y^2 - 8x + 2y + 7 = 0, we substitute the coordinates of the point into the equation and check if it satisfies the equation.
Step 1: Substituting the coordinates into the equation:
(3)^2 + (2)^2 - 8(3) + 2(2) + 7 = 0
9 + 4 - 24 + 4 + 7 = 0
16 - 24 + 11 = 0
-8 + 11 = 0
3 = 0
Since the equation is not satisfied, the point (3,2) does not lie on the circle x^2 + y^2 - 8x + 2y + 7 = 0.
Finding the equation of the tangent at this point requires a different approach, since the point given does not lie on the circle. Could you provide the correct coordinates so we can proceed with finding the equation of the tangent?
Question 2:
To find the length of the tangents from the point (5,7) to the circle x^2 + y^2 - 4x - 6y + 9 = 0, we can use the tangent-secant theorem.
Step 1: Substitute the coordinates of the point (5,7) into the equation of the circle:
(5)^2 + (7)^2 - 4(5) - 6(7) + 9 = 0
25 + 49 - 20 - 42 + 9 = 0
74 - 62 = 0
12 = 0
Since the equation is not satisfied, the point (5,7) does not lie on the circle x^2 + y^2 - 4x - 6y + 9 = 0. Therefore, there are no tangents from this point to the circle.
Question 3:
To find the equation of the common chord of the circles x^2 + y^2 - 6x + 2y + 1 = 0 and x^2 + y^2 + 4x - 2y - 13 = 0, we can use the properties of chords intersecting inside the circles at right angles.
Step 1: Write the equations of the circles in general form:
Circle 1: x^2 + y^2 - 6x + 2y + 1 = 0
Circle 2: x^2 + y^2 + 4x - 2y - 13 = 0
Step 2: Multiply the equations by appropriate values to make the coefficients of x^2 and y^2 the same:
4*(Circle 1) => 4x^2 + 4y^2 - 24x + 8y + 4 = 0
-6*(Circle 2) => -6x^2 - 6y^2 - 24x + 12y + 78 = 0
Step 3: Add the two equations:
4x^2 + 4y^2 - 24x + 8y + 4 - 6x^2 - 6y^2 - 24x + 12y + 78 = 0
Simplifying the equation:
-2x^2 - 2y^2 - 48x + 20y + 82 = 0
The resultant equation, -2x^2 - 2y^2 - 48x + 20y + 82 = 0, represents the common chord of the two circles.