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:(a) (-6,-7)(b) (-10,-15)(c) (-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

Approved