Question

The angle between two tangents to the parabola y^ 2 = 4 a x is
constant and equal to a . Prove that the locus of their point of
intersection is given by y^2 - 4ax = (a + x)^2 tan^2 a.
What will be the locus if (i) a =
π/ 2
(ii) a = 45°
​

Answer 1

Answer:

Step-by-step explanation:

Parabola: y  

2

=4ax.......(i)

Let the point of intersection be (h,k)

Tangent to (i) be T:y=mx+  

m

a

​  

.......(ii)

(h,k) lies on T=>m  

2

h−mk+a=0........(iii)(quadraticinm)

=>m  

1

​  

+m  

2

​  

=  

h

k

​  

,m  

1

​  

m  

1

​  

=  

h

a

​  

,m  

1

​  

−m  

2

​  

=  

h

1

​  

 

k  

2

−4ah

​  

 

So let the angle between two tangents =α

=>tanα=±  

1+m  

1

​  

m  

2

​  

 

m  

1

​  

−m  

2

​  

 

​  

=±  

h+a

k  

2

−4ah

​  

 

​  

 

=>k  

2

−4ah=tan  

2

α(h+a)  

2

 

Locus: y  

2

−4ax=tan  

2

α(x+a)  

2

..........(iv)

As given α=45  

°

=>tanα=1

So, equation (iv) becomes y  

2

−4ax=(1)  

2

(x+a)  

2

 

=>y  

2

−4ax=(x+a)  

2

 is locus of point of intersection of tangents with angle between them equal to 45  

°

.