What is tangent of the circle ?
If the line touches the circle exactly at one point, then it is called tangent.
For a line and a circle, there may be the following three cases available.
Problem 1 :
Show that the line y = -3x - 10 is the tangent to the circle
x^{2} + y^{2} - 8x + 4y - 20 = 0
and also find the point of contact.
Solution :
x^{2} + y^{2} - 8x + 4y - 20 = 0 ------(1)
y= -3x - 10 ----(2)
Applying the value of y in (1), we get
x^{2} + (-3x - 10)^{2} - 8x + 4(-3x - 10) - 20 = 0
x^{2} + 9x^{2} + 60x + 100 - 8x - 12x - 40 - 20 = 0
10x^{2} + 40x + 40 = 0
x^{2} + 4x + 4 = 0
(x + 2)(x + 2) = 0
x = -2 and x = -2
In both case, we have two same values for x. It shows the line touches the curve at one point. So it must be tangent.
Finding point of contact :
When x = -2,
y = -3(-2) - 10
y = -16
So, the point of contact is (-2, -16).
Problem 2 :
The circle
x^{2} + y^{2} + 4x - 7y - 8 = 0
cuts the y-axis at two points. Find the coordinates of these points.
Solution :
x^{2} + y^{2} + 4x - 7y - 8 = 0
Since the circle cuts the y-axis at two points, at point of intersection the value of x will be 0.
If x = 0
y^{2} - 7y - 8 = 0
(y - 8)(y + 1) = 0
y = 8 and y = -1
So, the points of contact are (0, 8) and (0, -1).
Problem 3 :
The circle
x^{2} + y^{2} - 2x + 10y - 24 = 0
cuts the x-axis at the points A and B. Find the length of AB.
Solution :
Since the circle cuts the x-axis at two points, at point of intersection the value of y will be 0.
If y = 0
x^{2} - 2x - 24 = 0
(x - 6) (x + 4) = 0
x = 6 and x = -4
So, the points of intersection are A(6, 0) and B(-4, 0).
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM