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  • The tangents at P and Q on the parabola   meet in T, S is the focus, SP, ST and

The tangents at P and Q on the parabola \mathrm{y^2=4 x}  meet in T, S is the focus, SP, ST and SQ are equal to a, b, and c respectively. Then the roots of the equation \mathrm{a x^2+2 b x+c=0} are

Option: 1

real and different


Option: 2

real and equal


Option: 3

complex roots


Option: 4

none of these 

 


Answers (1)

best_answer

\mathrm{\text { If } \mathrm{P}\left(\mathrm{t}_1\right), \mathrm{Q}\left(\mathrm{t}_2\right) \text { then coordinates of }}

\mathrm{\begin{gathered} \mathrm{T}=\left(\mathrm{t}_1 \mathrm{t}_2, \mathrm{t}_1+\mathrm{t}_2\right) \quad \mathrm{S}=(1,0) \\ \mathrm{a}=\mathrm{SP}=\left(1+\mathrm{t}_1{ }^2\right) \text { and } \mathrm{c}=\mathrm{SQ}=\left(1+\mathrm{t}_2{ }^2\right) \end{gathered}}

\mathrm{\begin{aligned} \mathrm{b}^2 & =\mathrm{ST}^2=\left(\mathrm{t}_1 \mathrm{t}_2-1\right)^2+\left(\mathrm{t}_1+\mathrm{t}_2\right)^2 \\ & =\mathrm{t}_1{ }^2+\mathrm{t}_2{ }^2+1+\mathrm{t}_1{ }^2 \mathrm{t}_2{ }^2=\left(1+\mathrm{t}_1{ }^2\right)\left(1+\mathrm{t}_2{ }^2\right) \end{aligned}}

\mathrm{\begin{aligned} & b^2=a c \Rightarrow \text { discriminant of the equation } 4 b^2-4 a c=0 \\ & b^2=a c \end{aligned}}

Hence roots are real and equal.

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chirag

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