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Let a line y=mx(m> 0) intersect the parabola, y^{2}=x at a point P, oyher than the origin. Let the tangent to it at P meet the x-axis at the point Q. If area (\Delta OPQ)=4\; sq.units, then m is equal to _____.
Option: 1 2  
Option: 21
Option: 3 0.5  
Option: 4 4
 

Answers (1)

best_answer

 

 

Area of Triangle -

Area of Triangle

If vertices of a triangle ABC given as A (x1, y1), B (x2, y2) and C(x3, y3) then area of ΔABC is 

\\\mathrm{\frac{1}{2}\begin{vmatrix} x_1 &y_1 &1 \\x_2 &y_2 &1 \\x_3 &y_3 &1 \end{vmatrix}=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|}

Area of an n-sided polygon  

The area of a polygon whose vertices are (x1, y1), (x2, y2), (x3, y3)............(xn, yn) taken in cyclic order is given by

=\frac{1}{2}\left | \left \{ \left ( x_1y_2-x_2y_1 \right )+\left ( x_2y_3-x_3y_2 \right )+\ldots\ldots+\left ( x_ny_1-x_1y_n \right ) \right \} \right |

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Tangents of Parabola in Parametric Form -

Tangents of Parabola in Parametric Form

\\ {\text { The equation of tangent to the parabola } \mathrm{y}^{2}=4 \mathrm{ax} \text { at the point }\left(\mathrm{at}^{2}, 2 \mathrm{at}\right) \text { is }} \\ {\mathrm{ty}=\mathrm{x}+\mathrm{at}^{2}}

\\ {\text { Equation of the tangent to the parabola } \mathrm{y}^{2}=4 \mathrm{ax} \text { at the point } \mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \text { is }} \\ {\mathrm{yy}_{1}=2 \mathrm{a}\left(\mathrm{x}+\mathrm{x}_{1}\right)} \\ {\text { replace } \mathrm{x}_{1} \rightarrow \mathrm{at}^{2}, \mathrm{y}_{1} \rightarrow 2 \mathrm{at}} \\ {\mathrm{y}(2 \mathrm{at})=2 \mathrm{a}\left(\mathrm{x}+\mathrm{at}^{2}\right) \Rightarrow \mathrm{yt}=\mathrm{x}+\mathrm{at}^{2}}

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\begin{array}{l}{2 \mathrm{ty}=\mathrm{x}+\mathrm{t}^{2}} \\ {\mathrm{Q}\left(-\mathrm{t}^{2}, 0\right)}\end{array}

\\\frac{1}{2}\begin{vmatrix} 0 &0 &1 \\ t^2 & t & 1\\ -t^2& 0 &1 \end{vmatrix}=4\\t=\pm2\\m=1/2=0.5

Correct Option (3)

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vishal kumar

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