Tangents are drawn from the point `(x_1,y_1)` to the parabola `y^2=4ax` show that the length of their chord of contact is `1/|a|sqrt((y_1^2-4ax_1)(y_1^2+4a^2))`.

Options

A. `1/asqrt((y_(1)^(2)-4ax_(1))(y_(1)^(2)+4a^(2)))`
B. `sqrt((y_(1)^(2)-4ax_(1))(y_(1)^(2)+4a^(2)))`
C. `1/asqrt((y_(1)^(2)+4ax_(1))(y_(1)^(2)+4a^(2)))`
D. none of these

Answer

A

Explanation

Let QR be the chord of contact of tangents drawn from a point `P(x_(1), y_(1))` to the parabola `y^(2)=4ax`. Then, the equation of QR is
`yy_(1)=2a(x+x_(1))`
The ordinates of Q and R are the roots of the equation
`y^(2)=4a((yy_(1)-2ax_(1))/(2a))or, y^(2)-2yy_(1)+4ax_(1)=0`
`:." "k_(1)+k_(2)=2y_(1)" and "k_(1)k_(2)=4ax_(1)" ...(i)"`
Since `(h_(1),k_(1))" and "(h_(2),k_(2))` lie on the parabola `y^(2)=4ax`.

`:." "k_(1)^(2)=4ah_(1)" and "k_(2)^(2)=4ah_(2)`
`rArr" "k_(1)^(2)-k_(2)^(2)=4a(h_(1)-h_(2))rArr(k_(2)^(2)-k_(1)^(2))/(4a)=h_(2)-h_(1)" ....(ii)"`
`:." "QR=sqrt((k_(2)-k_(1))^(2)+(h_(2)-h_(1))^(2))`
`rArr" "QR=sqrt((k_(2)-k_(1))^(2)+((K_(2)^(2)-k_(1)^(2))^(2))/(16a^(2)))" "["Using (ii)"]`
`rArr" "QR=((k_(2)-k_(1)))/(4a)sqrt(16a^(2)+(k_(2)+k_(1)))`
`rArr" "QR=sqrt((k_(2)+k_(1)^(2))-4k_(1)k_(2))/(4a)xxsqrt(16a^(2)+(k_(2)+k_(1)))`
`rArr" "QR=sqrt(4y_(1)^(2)-16ax_(1))/(4a)xxsqrt(16a^(2)+4y_(1)^(2))`
`rArr" "QR=1/asqrt((y_(1)^(2)-4ax_(1))(y_(1)^(2)+4a^(2)))`