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  • Tangent and normal are drawn at  on the parabola <img alt="y^2=16 x" src="https://entrancecorner

Tangent and normal are drawn at P(16,16) on the parabola y^2=16 x which intersect the axis of the parabola at A and B, respectively. If C is the center of the circle through the points P, A and B and ∠ = CPB θ, then a value of tanθ is

Option: 1

\begin{aligned} & \frac{1}{2} \\ \end{aligned}


Option: 2

-\frac{10}{9} \\


Option: 3

3 \\


Option: 4

\frac{4}{3}


Answers (1)

best_answer

Equation of tangent and normal to the curve y^2=16 x \text { at } P(16,16) \text { is } x-2 y+16=0  and  2 x+y-48=0 .

A=(-16,0) ; B=(24,0)

\because C is the center of circle passing through PAB i.e.c(4,0)

Slope of   \begin{aligned} & P C=\frac{16-0}{16-4}=\frac{4}{3}=m_1 \\ \end{aligned}

Slope of   P B=\frac{16-0}{16-24}=\frac{16}{8}=-2=m_2

\begin{aligned} & \tan \theta=\frac{m_1-m_2}{1+m_1 m_2} \\ & \Rightarrow \tan \theta=\frac{\frac{4}{3}+2}{1-\frac{4}{3}(2)} \\ & \Rightarrow \tan \theta=-\frac{10}{9} \end{aligned}

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Pankaj

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