SATHEE: Parabola 2 Question 6

Parabola 2 Question 6

6. 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 centre of the circle through the points $P, A$ and $B$ and $∠ C P B = θ$ , then a value of $\tan θ$ is

(2018 Main)

(a) $\frac{1}{2}$

(b) 2

(d) $\frac{4}{3}$

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Answer:

Correct Answer: 6. (c)

Solution:

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

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

$∵ C$ is the centre of circle passing through $P A B$

i.e.

$C = (4, 0)$

Slope of $P C = \frac{16 - 0}{16 - 4} = \frac{16}{12} = \frac{4}{3} = m_{1}$

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

$\tan θ = \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}$

$\Rightarrow \tan θ = \frac{\frac{4}{3} + 2}{1 - \frac{4}{3} (2)} \Rightarrow \tan θ = 2$

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Parabola 2 Question 6

6. Tangent and normal are drawn at P(16,16) on the parabola y2=16x, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P,A and B and ∠CPB=θ, then a value of tan⁡θ is

Answer:

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6. 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 centre of the circle through the points $P, A$ and $B$ and $∠ C P B = θ$ , then a value of $\tan θ$ is