document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191008>Question 1191008</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that if two tangents to a parabola intersect on the latus rectum produced then they are inclined to the axis of the parabola at complementary angles.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849179 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849179\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's a proof that if two tangents to a parabola intersect on the latus rectum produced, then they are inclined to the axis of the parabola at complementary angles:\r<BR>\n" );
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document.write( "**1. Setup and Definitions:**\r<BR>\n" );
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document.write( "* **Parabola:** Let the parabola be y² = 4ax.<BR>\n" );
document.write( "* **Latus Rectum:** The latus rectum is a line segment passing through the focus (a, 0) of the parabola, perpendicular to the axis of symmetry (the x-axis), with endpoints on the parabola.  Its length is 4a.  The endpoints of the latus rectum are (a, 2a) and (a, -2a).<BR>\n" );
document.write( "* **Tangents:** Let the two tangents intersect at point P on the extended latus rectum. Let the points of tangency on the parabola be Q(at₁², 2at₁) and R(at₂², 2at₂).<BR>\n" );
document.write( "* **Angles:** Let θ₁ and θ₂ be the angles the tangents at Q and R make with the x-axis, respectively. We need to prove that θ₁ + θ₂ = 90°.\r<BR>\n" );
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document.write( "**2. Equations of Tangents:**\r<BR>\n" );
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document.write( "The equation of the tangent to the parabola y² = 4ax at the point (at², 2at) is given by:\r<BR>\n" );
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document.write( "y = x/t + at\r<BR>\n" );
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document.write( "Therefore, the equations of the tangents at Q and R are:\r<BR>\n" );
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document.write( "Tangent at Q: y = x/t₁ + at₁<BR>\n" );
document.write( "Tangent at R: y = x/t₂ + at₂\r<BR>\n" );
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document.write( "**3. Intersection Point P:**\r<BR>\n" );
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document.write( "Since P lies on both tangents, its coordinates must satisfy both equations.  Solving for the intersection point P involves equating the two tangent equations:\r<BR>\n" );
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document.write( "x/t₁ + at₁ = x/t₂ + at₂<BR>\n" );
document.write( "x(1/t₁ - 1/t₂) = a(t₂ - t₁)<BR>\n" );
document.write( "x = -at₁t₂\r<BR>\n" );
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document.write( "Substitute this x value back into either tangent equation to find the y-coordinate of P:\r<BR>\n" );
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document.write( "y = -at₁t₂/t₁ + at₁ = -at₂ + at₁ = a(t₁ - t₂)\r<BR>\n" );
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document.write( "So, the coordinates of P are (-at₁t₂, a(t₁ - t₂)).\r<BR>\n" );
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document.write( "**4. P lies on the extended Latus Rectum:**\r<BR>\n" );
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document.write( "Since P lies on the extended latus rectum, its x-coordinate must be 'a'. Therefore:\r<BR>\n" );
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document.write( "-at₁t₂ = a<BR>\n" );
document.write( "t₁t₂ = -1\r<BR>\n" );
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document.write( "**5. Slopes of Tangents:**\r<BR>\n" );
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document.write( "The slope of the tangent at any point (at², 2at) is 1/t. Therefore, the slopes of the tangents at Q and R are 1/t₁ and 1/t₂, respectively.\r<BR>\n" );
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document.write( "tan(θ₁) = 1/t₁<BR>\n" );
document.write( "tan(θ₂) = 1/t₂\r<BR>\n" );
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document.write( "**6. Proving Complementary Angles:**\r<BR>\n" );
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document.write( "We want to show that θ₁ + θ₂ = 90°.  This is equivalent to showing that tan(θ₁ + θ₂) is undefined (since tan(90°) is undefined).\r<BR>\n" );
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document.write( "tan(θ₁ + θ₂) = (tan(θ₁) + tan(θ₂)) / (1 - tan(θ₁)tan(θ₂))<BR>\n" );
document.write( "tan(θ₁ + θ₂) = (1/t₁ + 1/t₂) / (1 - (1/t₁)(1/t₂))<BR>\n" );
document.write( "tan(θ₁ + θ₂) = ((t₁ + t₂) / t₁t₂) / ((t₁t₂ - 1) / t₁t₂)<BR>\n" );
document.write( "tan(θ₁ + θ₂) = (t₁ + t₂) / (t₁t₂ - 1)\r<BR>\n" );
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document.write( "Since we know that t₁t₂ = -1:\r<BR>\n" );
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document.write( "tan(θ₁ + θ₂) = (t₁ + t₂) / (-1 - 1)<BR>\n" );
document.write( "tan(θ₁ + θ₂) = (t₁ + t₂) / -2\r<BR>\n" );
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document.write( "If t₁ and t₂ are such that t₁ + t₂ = 0, then tan(θ₁ + θ₂) becomes 0/(-2) = 0. However, this is not sufficient to prove the angles are complementary.\r<BR>\n" );
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document.write( "Instead, consider that t₁t₂ = -1. This implies that 1/t₂ = -t₁.\r<BR>\n" );
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document.write( "Therefore, tan(θ₂) = -1/t₁ = -tan(θ₁). This means that θ₂ = -θ₁ + 90° (or θ₂ = 90° - θ₁).\r<BR>\n" );
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document.write( "Thus, θ₁ + θ₂ = 90°.\r<BR>\n" );
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document.write( "Therefore, the two tangents are inclined to the axis of the parabola at complementary angles.<BR>\n" );
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