CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

A variable circle passes through the fixed point A(p, q) and touches the x-axis. The locus of the other end of the diameter through A is


A
(xp)2=4qy
loader
B
(xq)2=4py
loader
C
(yp)2=4qx
loader
D
(yq)2=4px
loader

Solution

The correct option is D (xp)2=4qy
In a circle, AB is a diameter where the coordintes of A are (p, q). Let the coordinates of B be (x1,y1).
The equation of the circle in diameter form is 
(xp)(xx1)+(yq)(yy1)=0x2(p+x1)x+px1+y2(y1+q)y+qy1=0x2(p+x1)x+y2(y1+q)y+px1+qy1=0
Since this circle touches the x-axis, 
y=0
x2(p+x1)x+px1+qy1=0
Also the discriminant of the above equation will be equal to zero because the circle touches the x-axis. Therefore,
(p+x1)2=4(px1+qy1)p2+x21+2px1=4px1+4qy1x212px1+p2=4qy1
Therefore, the locus of the point B is 
(xp)2=4qy

Mathematics

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image