Find the locus of the point of intersection of tangents in the parabola `x^2=4a xdot` which are inclined at an angle `theta` to each other. Which intercept constant length `c` on the tangent at the vertex. such that the area of ` A B R` is constant `c ,` where `Aa n dB` are the points of intersection of tangents with the y-axis and `R` is a point of intersection of tangents.

Explanation

(i)
Equations of tangents at points P and Q are, respectively,
`t_(1)y=x+at_(1)^(2)andt_(2)y=x+at_(2)^(2)`.
These tangents intersect at R.
`:." "t_(1)t_(2)=(h)/(a)andt_(1)+t_(2)=(k)/(a)`
The angle between tangents is `theta`.
`:." "tantheta=|((1)/(t_(2))-(1)/(t_(2)))/(1+(1)/(t_(1))(1)/(t_(2)))|=|(t_(1)-t_(2))/(1+t_(1)t_(2))|`
`rArr" "tan^(2)theta(1+t_(1)t_(2))^(2)=(t_(1)-t_(2))^(2)`
`rArr" "tan^(2)theta(1+t_(1)t_(2))^(2)=(t_(1)+t_(2))^(2)+4t_(1)t_(2)`
`rArr" "tan^(2)theta(1+(h)/(a))^(2)=((k)/(a))^(2)+4(h)/(a)`
`rArr" "tan^(2)theta(x+a)^(2)=y^(2)+4ax`
(Replacing h by x and k be y)
This is the required equation of locus.
(ii) Tangents at P and Q meet y-axis at `A(0,at_(1))andB(0,at_(2))`, respectively.
Given that AB = c
`:." "|at_(1)-at_(2)|=c`
`rArr" "a^(2)[(t_(1)+t_(2))^(2)-4t_(1)t_(2)]=c^(2)`
`rArr" "a^(2)[((k)/(a))^(2)-4(h)/(a)]=c^(2)`
`rArr" "y^(2)-4ax=c^(2)`, which is the required equation of locus.
(iii) Area of triangle ABR = c
`rArr" "(1)/(2)AB*RM=c`
`rArr" "(1)/(2)|at_(1)-at_(2)|(at_(1)t_(2))=c^(2)`
`rArr" "a^(4)[(t_(1)+t_(2))-4t_(1)t_(2)](t_(1)t_(2))^(2)=4c^(2)`
`rArr" "a^(4)[((k)/(a))^(2)-4(h)/(a)]((h)/(a))^(2)=4c^(2)`
`rArr" "(y^(2)-4ax)x^(2)=4c^(2)`, which is the required equation of locus.