Question

From the point m(1,3) a tangent is drawn at point p to the parabola 2{(x-6 )2 + (y-6)2}=(x+y-4)2. find the measure of anglemsp where s is the focus of the parabola

Answer 1

Thank you for asking this question. Here is your answer:

2{(x-6 )2 + (y-6)2}=(x+y-4)2 (this is not a defined parabola as it consists both x and y variables with power 2)

Only one variable either x or y should have power 2

x=-a

y=a(t^2-1)/t

slope msp= (2at-0)/(at^2-a)=(2t)/(t^2-1)

slope msm= -(t^2-1)/2t

msp×msm= -1

hence angle is 90°

If there is any confusion please leave a comment below.

Answer 2

"the angle msp is independent of coordinate system hence we shift the parabola

2{(x-6)^2+(y-6)^2}=(x+y-4)^2 parallel to y axis the angle msp never change hence the equation of parabola becomes y^2=4ax

point p(at^2,2at)

equation of tangent to the parabola y^2=4ax is

ty=x+at^2  ---(1)

dirctrix cut axis at(-a,0)

putting x in point(-a,0) in eqn(1) we get coordinates of m

x=-a

y=a(t^2-1)/t

slope msp= (2at-0)/(at^2-a)=(2t)/(t^2-1)

slope msm= -(t^2-1)/2t

msp×msm= -1

hence angle is 90°

"