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If \mathrm{a, b, c}  are rational and the tangent to the parabola \mathrm{y^2=4 k x ~at P(p, a)} and \mathrm{Q(q, b)}  meet at \mathrm{R(r, c)}, then the equation \mathrm{a x^2+b x-2 c=0} has
 

Option: 1

 imaginary roots
 


Option: 2

 real and equal roots
 


Option: 3

 rational roots


Option: 4

irrational roots


Answers (1)

best_answer

Given parabola is \mathrm{y^2=4 k x} \quad \quad \dots(i)

Let \mathrm{P\equiv\left(k t_1^2, 2 k t_1\right), Q \equiv\left(k t_2^2, 2 k t_2\right)}

Since tangents at P and Q meet at R

\mathrm{\therefore \quad R \equiv\left[k t_1 t_2, k\left(t_1+t_2\right)\right] }

Given \mathrm{2 k t_1=a, 2 k t_2=b, k\left(t_1+t_2\right)=c }

\begin{aligned} & \Rightarrow \mathrm{a+b-2 c=0 }\\ & \Rightarrow 1 \text { is a root of equation } \mathrm{a x^2+b x-2 c=0 .} \end{aligned}

But 1 is a rational number and a, b, c are rational.
\therefore Both roots of equation \mathrm{a x^2+b x-2 c=0}  are rational.

Posted by

Ritika Jonwal

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