A tangent to the parabola [tex]$y^2=4 a x$[/tex] makes an angle of [tex]$60^{\circ}$[/tex] with the [tex]$x$[/tex]-axis; find its point to contact.
Answer (1)
Find the slope of the tangent using the angle it makes with the x-axis: m = tan ( 6 0 ∘ ) = 3 .
Use the formula for the tangent to a parabola y 2 = 4 a x , which is y = m x + m a .
Determine the point of contact using the formula ( m 2 a , m 2 a ) .
Substitute m = 3 to find the point of contact: ( 3 a , 3 2 a 3 ) .
Explanation
Problem Analysis We are given a parabola y 2 = 4 a x and a tangent to it that makes an angle of 6 0 ∘ with the x-axis. Our goal is to find the point of contact of this tangent with the parabola.
Finding the Slope The slope of the tangent is given by the tangent of the angle it makes with the x-axis. Therefore, the slope m is tan ( 6 0 ∘ ) . We know that tan ( 6 0 ∘ ) = 3 . So, m = 3 .
Equation of the Tangent The equation of the tangent to the parabola y 2 = 4 a x with slope m is given by y = m x + m a . Substituting m = 3 , we get y = 3 x + 3 a .
Finding the Point of Contact The point of contact of the tangent to the parabola is given by the coordinates ( m 2 a , m 2 a ) . Substituting m = 3 , we get the point of contact as ( ( 3 ) 2 a , 3 2 a ) = ( 3 a , 3 2 a ) .
Rationalizing the Denominator To rationalize the denominator of the y-coordinate, we multiply the numerator and denominator by 3 : 3 2 a = 3 2 a 3 . Therefore, the point of contact is ( 3 a , 3 2 a 3 ) .
Final Answer The point of contact of the tangent to the parabola y 2 = 4 a x that makes an angle of 6 0 ∘ with the x-axis is ( 3 a , 3 2 a 3 ) .
Examples
Imagine you are designing a parabolic reflector for a solar oven. Knowing the point of contact of a tangent line at a specific angle helps you precisely focus sunlight onto a single point, maximizing the oven's efficiency. This ensures optimal cooking performance by accurately directing solar energy.
