Question

A tangent to the parabola

y 2=12x makes an angle of 45° with the straight line
x−2y+3=0. Find its equation and the point of contact.

a) y=2x−3 and P(3,6)
b) y=2x+3 and P(−3,−6)
c) y=−2x−3 and P(3,−6)
d) y=−2x+3 and P(−3,6)

Answer 1

To find the equation of the tangent to the parabola and the point of contact, we need to find the coordinates of the point of contact and the slope of the tangent. The correct answer is option (b) y = 2x + 3 and P(-3, -6).

Step-by-step explanation:

To find the equation of the tangent to the parabola and the point of contact, we need to find the coordinates of the point of contact and the slope of the tangent.

We are given that the tangent makes an angle of 45 degrees with the line x - 2y + 3 = 0. The line x - 2y + 3 = 0 can be written in the form y = (x + 3)/2. To find the slope of the line, we can take the derivative of y with respect to x and evaluate it at the point of contact.

The derivative of y = (x + 3)/2 is dy/dx = 1/2. Since the tangent makes an angle of 45 degrees with the line, the slope of the tangent is tan(45) = 1. Therefore, the equation of the tangent is y - y1 = m(x - x1), where (x1, y1) is the point of contact. Substituting the slope and the coordinates of the point of contact into the equation, we get y - y1 = 1(x - x1).

Since the tangent goes through the point (x1, y1), we can substitute these coordinates into the equation to find the equation of the tangent.

The correct answer is option (b) y = 2x + 3 and P(-3, -6).