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Locus of point of intersection of tangents

  • #1
utkarshakash
Gold Member
855
13

Homework Statement


Locus of the point of intersection of tangents to the parabolas y[itex]^{2}[/itex]=4(x+1) and y[itex]^{2}[/itex]=8(x+2) which are at right angles, is


Homework Equations


Equation of tangent for first parabola
t[itex]_{1}[/itex]y=x+1+at[itex]_{1}[/itex][itex]^{2}[/itex]
Equation of tangent for second parabola
t[itex]_{2}[/itex]y=x+2+bt[itex]_{2}[/itex][itex]^{2}[/itex]

The Attempt at a Solution


Let us assume that the point of intersection of tangents is (h,k)
Since it lies on the tangent
∴kt[itex]_{1}[/itex]=h+1+at[itex]_{1}[/itex][itex]^{2}[/itex]
[itex]\Rightarrow[/itex]at[itex]_{1}[/itex][itex]^{2}[/itex]-kt[itex]_{1}[/itex]+(h+1)=0
t[itex]_{1}[/itex]t[itex]_{2}[/itex]=h+1 (product of roots)
Also t[itex]_{1}[/itex]t[itex]_{2}[/itex]=-1 (Since they are at right angles)
∴required locus = x+2=0

I can't understand where I'm wrong. Is the equation of tangent incorrect? Please Help.
 

Answers and Replies

  • #2
ehild
Homework Helper
15,396
1,803

Homework Statement


Locus of the point of intersection of tangents to the parabolas y[itex]^{2}[/itex]=4(x+1) and y[itex]^{2}[/itex]=8(x+2) which are at right angles, is


Homework Equations


Equation of tangent for first parabola
t[itex]_{1}[/itex]y=x+1+at[itex]_{1}[/itex][itex]^{2}[/itex]
Equation of tangent for second parabola
t[itex]_{2}[/itex]y=x+2+bt[itex]_{2}[/itex][itex]^{2}[/itex]

The Attempt at a Solution


Let us assume that the point of intersection of tangents is (h,k)
Since it lies on the tangent
∴kt[itex]_{1}[/itex]=h+1+at[itex]_{1}[/itex][itex]^{2}[/itex]
[itex]\Rightarrow[/itex]at[itex]_{1}[/itex][itex]^{2}[/itex]-kt[itex]_{1}[/itex]+(h+1)=0
t[itex]_{1}[/itex]t[itex]_{2}[/itex]=h+1 (product of roots)
Also t[itex]_{1}[/itex]t[itex]_{2}[/itex]=-1 (Since they are at right angles)
∴required locus = x+2=0

I can't understand where I'm wrong. Is the equation of tangent incorrect? Please Help.
I can not follow your derivation. What have you got for a and b to make the equations tangent to the parabolas? The product of roots in the equation in red is (h+1)/a. t2 is not the other root of the equation for the first tangent line. Use the equations of both tangent lines to get the intersection.

I also got x=const. for the locus, but different from yours.

ehild
 
Last edited:
  • #3
utkarshakash
Gold Member
855
13
I can not follow your derivation. What have you got for a and b to make the equations tangent to the parabolas? The product of roots in the equation in red is (h+1)/a. t2 is not the other root of the equation for the first tangent line. Use the equations of both tangent lines to get the intersection.

I also got x=const. for the locus, but different from yours.

ehild
Hey actually i arrived at the wrong answer because I only considered the first equation but not second. By solving both I get x+2=0 and x+4=0. Adding these two yields 2x+6=0 or x+3=0 which is the required locus.

P.S.- I wrote product of roots as h+1 because a=1. so (h+1)/a=(h+1)/1=(h+1) :wink:
 
  • #4
ehild
Homework Helper
15,396
1,803
Hey actually i arrived at the wrong answer because I only considered the first equation but not second. By solving both I get x+2=0 and x+4=0. Adding these two yields 2x+6=0 or x+3=0 which is the required locus.
x+2=0 and x+4=0 contradict to each other, so that is not the right way to go, but the end result is correct. I got the same .

P.S.- I wrote product of roots as h+1 because a=1. so (h+1)/a=(h+1)/1=(h+1) :wink:
But you kept that a=1 top secret. It took me some time to find out what your equations mean.

ehild
 
  • #5
ehild
Homework Helper
15,396
1,803
As it is a nice problem, and the OP has solved it, I write out the steps in a more ordered form.

We have two parabolas

(1)[tex]y^2=4(x+1)[/tex]
(2)[tex]y^2=8(x+2)[/tex]

The tangent lines are
[tex]x=ty+q[/tex].
Find q in terms of t so the line is tangent to the parabola, that is, the line and the parabola have a single common point. Substitute for x into the equation of parabola(1):
(1)
[tex]y^2-4t_1y-4(q_1+1)=0[/tex]
The quadratic equation has a single solution, so the discriminant is zero--> [tex]D=16t_1^2+16(q_1+1)=0\rightarrow q_1=-(t_1^2+1)[/tex]
(2)Substitute the equation of the tangent line for x in the equation of the second parabola: [tex]y^2-8t_21y-8(q_1+2)=0[/tex]
The discriminant is [tex]D=64t_2^2+32(q_2+2)=0\rightarrow q_2=-2(t_1^2+1)[/tex]

The tangent lines are perpendicular so [tex]t_1t_2=-1[/tex]
Let t1=t, t2=-1/t. With that notation:
(1)[tex]x=ty-(t^2+1) \rightarrow y=x/t+(t+1/t)[/tex]
(2)[tex]x=-y/t-2(-1/t^2+1) \rightarrow y=-tx-2(t+1/t)[/tex]
Subtract the expressions for y to get the abscissa of the locus.

ehild
 
  • #6
utkarshakash
Gold Member
855
13
As it is a nice problem, and the OP has solved it, I write out the steps in a more ordered form.

We have two parabolas

(1)[tex]y^2=4(x+1)[/tex]
(2)[tex]y^2=8(x+2)[/tex]

The tangent lines are
[tex]x=ty+q[/tex].
Find q in terms of t so the line is tangent to the parabola, that is, the line and the parabola have a single common point. Substitute for x into the equation of parabola(1):
(1)
[tex]y^2-4t_1y-4(q_1+1)=0[/tex]
The quadratic equation has a single solution, so the discriminant is zero--> [tex]D=16t_1^2+16(q_1+1)=0\rightarrow q_1=-(t_1^2+1)[/tex]
(2)Substitute the equation of the tangent line for x in the equation of the second parabola: [tex]y^2-8t_21y-8(q_1+2)=0[/tex]
The discriminant is [tex]D=64t_2^2+32(q_2+2)=0\rightarrow q_2=-2(t_1^2+1)[/tex]

The tangent lines are perpendicular so [tex]t_1t_2=-1[/tex]
Let t1=t, t2=-1/t. With that notation:
(1)[tex]x=ty-(t^2+1) \rightarrow y=x/t+(t+1/t)[/tex]
(2)[tex]x=-y/t-2(-1/t^2+1) \rightarrow y=-tx-2(t+1/t)[/tex]
Subtract the expressions for y to get the abscissa of the locus.

ehild
Thanks for helping.
 

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