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Two parabolas \mathrm{y^2=4 a\left(x-\lambda_1\right)} and \mathrm{x^2=4 a\left(y-\lambda_2\right)} touch each other for all values of parameters \mathrm{\lambda_1, \lambda_2}. Their point of contact lie on

Option: 1

a straight line
 


Option: 2

a circle
 


Option: 3

a parabola
 


Option: 4

a hyperbola


Answers (1)

best_answer

Let the point of contact be \mathrm{P\left(x_1, y_1\right).}

The tangent at P to the parabolas

\mathrm{ y^2=4 a\left(x-\lambda_1\right) }

and  \mathrm{x^2=4 a\left(y-\lambda_2\right)} are
\mathrm{ \begin{array}{ll} & y y_1=2 a\left(x+x_1\right)-4 a \lambda_1 \\\\ \Rightarrow \quad & 2 a x-y y_1=2 a\left(2 \lambda_1-x_1\right)\, \, .....(1) \\\\ \text { And } & x x_1=2 a\left(y+y_1\right)-4 a \lambda_2 \\\\ \Rightarrow \quad & x x_1-2 a y=2 a\left(y_1-2 \lambda_2\right)\, \, .....(2) \end{array} }

(1) and (2) represent the same tangent. On comparing the coefficients in (1) and (2),

\mathrm{ \frac{2 a}{x_1}=\frac{y_1}{2 a} \Rightarrow x_1 y_1=4 a^2 }
The locus of \mathrm{P\left(x_1, y_1\right)} is

\mathrm{ x y=4 a^2 }

which is a rectangular hyperbola.

The answer is (d)

Posted by

Suraj Bhandari

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