Exams Joint Entrance Examination Mathematics Parabola Question 53 Question 53 of 198 Medium NUMERICAL
The ordinates of the points P and Q Q Q on the parabola with focus ( 3 , 0 ) (3,0) ( 3 , 0 ) and directrix x = − 3 x=-3 x = − 3 are in the ratio 3 : 1 3:1 3 : 1 . If R ( α , β ) R(\alpha ,\beta ) R ( α , β ) is the point of intersection of the tangents to the parabola at P P P and Q Q Q , then β 2 α \frac{{\beta }^{2}}{\alpha } α β 2 is equal to \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.
Answer: 16
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y 2 = 12 x ( ∵ a = 3 ) P ≡ ( a t 2 1 , 2 a t 1 ) Q ≡ ( a t 2 2 , 2 a t 2 ) R ( α , β ) ≡ ( a t 1 t 2 , a ( t 1 + t 2 ) ) ≡ ( ( 3 t ) ( 3 t ) , 3 ( t + 3 t ) ) = ( 9 t 2 , 12 t ) ∴ β 2 α = 144 t 2 9 t 2 = 16 \begin{gathered} {y}^{2}=12x\, (\because a=3) \\ P\equiv (a{t}_{2}^{1},2a{t}_{1}) \\ Q\equiv (a{t}_{2}^{2},2a{t}_{2}) \\ R(\alpha ,\beta )\equiv (a{t}_{1}{t}_{2},a({t}_{1}+{t}_{2})) \\ \equiv ((3t)(3t),3(t+3t))=(9{t}^{2},12t) \\ \therefore \frac{{\beta }^{2}}{\alpha }=\frac{144{t}^{2}}{9{t}^{2}}=16 \end{gathered} y 2 = 12 x ( ∵ a = 3 ) P ≡ ( a t 2 1 , 2 a t 1 ) Q ≡ ( a t 2 2 , 2 a t 2 ) R ( α , β ) ≡ ( a t 1 t 2 , a ( t 1 + t 2 )) ≡ (( 3 t ) ( 3 t ) , 3 ( t + 3 t )) = ( 9 t 2 , 12 t ) ∴ α β 2 = 9 t 2 144 t 2 = 16 Answer the question to unlock