A tangent to the parabola `y^(2) = 8x` makes an angle of `45^(@)` with the straight line `y = 3x + 5`. Find its equation and its point of contact.

Explanation

Let the slope of the tangent be m
`thereforetan45^(@)=abs((3-m)/(1+3m))`
`!1+3m=pm(3-m)`
`thereforem=-2` or `1/2`
As we know that equation of tangent of slope m to the parabola `y^(2)=4ax` is `y=mx+a/m` and point of contact is `(a/(m^(2)),-(2a)/m)`
for m = –2, equation of tangent is y = –2x – 1 and point of contact is `(1/2,-2)`
for `m=1/2`, equation of tangent is `y=1/2x+4` and point of contact is (8, 8).