A tangent to the parabola [tex]y^2=12 x[/tex] makes an angle [tex]45^{\circ}[/tex] with the straight line [tex]x-2 y+3 \doteq 0[/tex]. Find its equation and the point of contact.
Answer (2)
Find the slope of the given line: m 1 = 2 1 .
Use the angle between two lines formula to find the slopes of the tangents: m = 3 and m = − 3 1 .
Use the tangent equation y = m x + m a to find the tangent equations: y = 3 x + 1 and y = − 3 1 x − 9 .
Find the points of contact using the formula ( m 2 a , m 2 a ) : ( 3 1 , 2 ) and ( 27 , − 18 ) .
y = 3 x + 1 , ( 3 1 , 2 ) ; y = − 3 1 x − 9 , ( 27 , − 18 )
Explanation
Problem Analysis First, let's identify the key information. We have a parabola y 2 = 12 x and a line x − 2 y + 3 = 0 . The tangent to the parabola makes a 4 5 ∘ angle with the given line. Our goal is to find the equation of this tangent and its point of contact with the parabola.
Find the slope of the given line The equation of the given line is x − 2 y + 3 = 0 . We can rewrite this in slope-intercept form as 2 y = x + 3 , so y = 2 1 x + 2 3 . Thus, the slope of the given line, m 1 , is 2 1 .
Apply the angle between two lines formula Let m be the slope of the tangent line. The angle θ between two lines with slopes m 1 and m is given by the formula: tan ( θ ) = 1 + m m 1 m − m 1
In our case, θ = 4 5 ∘ , so tan ( 4 5 ∘ ) = 1 . Plugging in m 1 = 2 1 , we get: 1 = 1 + m ( 2 1 ) m − 2 1
1 = 1 + 2 m m − 2 1
Solve for m We have two cases to consider:
Case 1: 1 + 2 m m − 2 1 = 1 m − 2 1 = 1 + 2 m m − 2 m = 1 + 2 1 2 m = 2 3 m = 3
Case 2: 1 + 2 m m − 2 1 = − 1 m − 2 1 = − 1 − 2 m m + 2 m = − 1 + 2 1 2 3 m = − 2 1 m = − 3 1
Find the tangent equations The general equation of a tangent to the parabola y 2 = 4 a x is y = m x + m a . In our case, y 2 = 12 x , so 4 a = 12 , which means a = 3 . Therefore, the equation of the tangent is y = m x + m 3 .
For m = 3 , the tangent equation is: y = 3 x + 3 3 = 3 x + 1
For m = − 3 1 , the tangent equation is: y = − 3 1 x + − 3 1 3 = − 3 1 x − 9
Find the points of contact The point of contact of the tangent y = m x + m a with the parabola y 2 = 4 a x is given by ( m 2 a , m 2 a ) . In our case, a = 3 , so the point of contact is ( m 2 3 , m 6 ) .
For m = 3 , the point of contact is: ( 3 2 3 , 3 6 ) = ( 9 3 , 2 ) = ( 3 1 , 2 )
For m = − 3 1 , the point of contact is: ( ( − 3 1 ) 2 3 , − 3 1 6 ) = ( 9 1 3 , − 18 ) = ( 27 , − 18 )
Final Answer Therefore, the equations of the tangents are y = 3 x + 1 and y = − 3 1 x − 9 , and the corresponding points of contact are ( 3 1 , 2 ) and ( 27 , − 18 ) .
Examples
Understanding tangents to parabolas is crucial in various fields, such as optics and engineering. For instance, in designing parabolic reflectors for satellite dishes or solar cookers, the tangent at a point determines the direction of reflected rays. By controlling the angle of the tangent, engineers can focus incoming signals or sunlight onto a specific point, optimizing the device's efficiency. This principle ensures that the energy is concentrated effectively, whether it's for receiving weak signals from space or harnessing solar power for cooking.
The tangents to the parabola y 2 = 12 x are y = 3 x + 1 at point ( 3 1 , 2 ) and y = − 3 1 x − 9 at point ( 27 , − 18 ) .
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