A variable circle is drawn to touch the line 3x - 4y = 10 and also the circle x2 + y2 = 1
externally then locus of centre is :
(A) straight line
circle
(C) pair of real, distinct straight lines (D) parabola
answer the question and I will follow you
Answers
Locus of the center of the circle :
16x² + 9y² + 24xy + 30x - 40y+25= 0
Let the center of the circle be (x1,y1).
The circle touches the line 3x - 4y = 10 , so the distance of the line from the center of the circle is equal to the radius of the circle.
radius = r = √(x1² + y1² - c)
Distance = d = l(3x1 - 4y1 -10)l / 5
=> r = d
The circle also touches the circle x² + y² = 1 .
The distance between the centers of the two circles is equal to the sum of the radius of the two circles.
=> √(((x1-0)²+(y1-0)²) = 1 + r
=> √(((x1-0)²+(y1-0)²) = 1 + d (given , d = r)
=> x1² + y1² = (1+d)²
=> x1² + y1² = (1 + (3x1 - 4y1 -10) / 5)²
=> x1² + y1² = ( (3x1 - 4y1 -5)/5)²
=> 25x1² + 25y1² = (3x1 - 4y1 -5)²
=> 25x1² + 25y1² = 9x1² + 16y1² - 24 x1y1 + 40y1 - 30x1
=> 16x1² + 9y1² + 24x1y1 + 30x1 - 40y1 + 25 = 0
Locus of the center of the circle :
16x² + 9y² + 24xy + 30x - 40y+25= 0