Answer to Question #143653 in Analytic Geometry for Sarita bartwal

Question #143653
If the tangents at two points of a parabola are at right angles, then show that they intersects at a point on the directrix.
1
Expert's answer
2020-11-16T19:26:17-0500

Let equation of the parabola is, y2=4axy^2=4ax

Tangent at any point (x1,y1)(x_1,y_1) is given by yy1=2a(x+x1)yy_1 = 2a(x+x_1)


Let any two points on the parabola be (at12,2at1),(at22,2at2)(at_1^2,2at_1),(at_2^2,2at_2)

Then equation of the tangent at (at12,2at1)(at_1^2,2at_1) is given by,

2at1y=2a(x+at12)2at_1y=2a(x+at_1^2)

t1y=(x+at12)    y=1t1(x+at12)t_1y=(x+at_1^2) \implies y=\frac{1}{t_1}(x+at_1^2)


Equation of the tangent at point (at22,2at2)(at_2^2,2at_2) is given by

2at2y=2a(x+at22)2at_2y=2a(x+at_2^2)

t2y=(x+at22)    y=1t2(x+at22)t_2y=(x+at_2^2) \implies y=\frac{1}{t_2}(x+at_2^2)


Since, these these tangents are perpendicular then,

1t11t2=1    1t2=t1\frac{1}{t_1}\frac{1}{t_2} = -1 \implies \frac{1}{t_2} = -t_1 (1)


Also these tangents intersect, then point of intersection is

1t1(x+at12)=1t2(x+at22)\frac{1}{t_1}(x+at_1^2)=\frac{1}{t_2}(x+at_2^2)

from equation (1), I can write,


(x+at12)=t12(x+a1t12)(x+at_1^2) = -t_1^2(x+a\frac{1}{t_1^2})


(1t12)x=a(1t12)(1-t_1^2)x = -a(1-t_1^2)


    x=a\implies x=-a

It is equation of the directrix.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment