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in Parabola by (65 points)
Show that the line x+y-2=0 touches the Parabola x^2=-8y. Find the co-ordinate of the point of contact.

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1 Answer

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by (120 points)

ANSWER: ( 4 , - 2 )

We know that ,

The tangency condition of parabola is 

c = a/m 

Given line → x + y - 2 = 0 

Here c = - 2 and m = -1 

Parabola → x2 = - 8y 

a = 2 

a/m = 2/-1 = -2 

Hence c = a/m 

\(\therefore\) x + y - 2 = 0 is a tangent to the parabola x2 = - 8y .

The equation of tangent at point ( x1 , y) to the parabola x2 = - 8y is 

xx1 = -4( y  +  y)

xx1 = -4y - 4y1

xx1 + 4y + 4y1 = 0 

 comparing the above equation with x + y - 2 = 0 

x = 4 = - 2y1

x= 4 

y1 = - 2 

\(\therefore\) The coordinates point of contact are ( 4 , - 2 ).

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