Question
The parabola $${y^2} = kx$$ makes an intercept of length 4 on the line $$x - 2y = 1.$$ Then $$k$$ is :
A.
$$\frac{{\sqrt {105} - 5}}{{10}}$$
B.
$$\frac{{5 - \sqrt {105} }}{{10}}$$
C.
$$\frac{{5 + \sqrt {105} }}{{10}}$$
D.
none of these
Answer :
$$\frac{{\sqrt {105} - 5}}{{10}}$$
Solution :
$$\eqalign{
& {\text{Solving }}x - 2y = 1,\,{y^2} = kx,\,{\text{ we get}}\, \cr
& {y^2} = k\left( {1 + 2y} \right){\text{ or }}{y^2} - 2ky - k = 0 \cr
& \therefore \,{y_1} + {y_2} = 2k,\,\,{y_1}.{y_2} = - k \cr
& \therefore 16 = {\left( {{x_1} - {x_2}} \right)^2} + {\left( {{y_1} - {y_2}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {\frac{{y_1^2}}{k} - \frac{{y_2^2}}{k}} \right)^2} + {\left( {{y_1} - {y_2}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {{y_1} - {y_2}} \right)^2}.\left\{ {\frac{{{{\left( {{y_1} + {y_2}} \right)}^2}}}{{{k^2}}} + 1} \right\} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left\{ {{{\left( {{y_1} + {y_2}} \right)}^2} - 4{y_1}.{y_2}} \right\}\left\{ {\frac{{4{k^2}}}{{{k^2}}} + 1} \right\} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 5\left\{ {4{k^2} + 4k} \right\} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 20{k^2} + 20k\,;\,\,\,\,\,\,\,\,\,\,\,\therefore \,5{k^2} + 5k - 4 = 0 \cr} $$