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Q) the tangent drawn to the parabola y= x.x + 6 at the point (1,7) touches the circle x.x + y.y +16x +12y + c = 0 at the point:ย
(a) (-6,-7)
(b) (-10,-15)
(c) ย (-9,-7)
(d) (13,7)
y = x2+6
dy/dx = slope = m = 2x
m=2 ( at point (1,7) )
eq of tangent at 1,7 : y-7 = (x-1)2
2x-y+5 = 0 ......................1
x2+y2+16x+12y+c = 0
now we have to make a tanget at point (x1,y1) which lies on circle
replace x2 by xx1 , y2 by yy1 & x by x+x1/2 , y by y+y1/2
eq of tangent = x(x1+8) + y(y1+6) + 8x1+6y1+c = 0 .....................2
eq of line 1 touches the circle so eq1 & 2 are equivalent so
ratios of cofficient of x,y is constant
x1+8/2 = y1+6/-1
x1+2y1+20 = 0 ................3
(x1,y1) will satisfy eq 1 so
2x1-y1+5=0 .....................4
solving 3 & 4 we get
(x1,y1) = (-6,-7)
approve my ans if u like it
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