SATHEE: Parabola 3 Question 7

Parabola 3 Question 7

7. The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola, whose

(2009)

(a) vertex is $\frac{2 a}{3}, 0$

(b) directrix is $x = 0$

(d) focus is $(a, 0)$

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Answer:

Correct Answer: 7. $(a, d)$

Solution:

Let $P (α, β)$ be any point on the locus. Equation of pair of tangents from $P (α, β)$ to the parabola $y^{2} = 4 a x$ is

$[β y - 2 a (x + α)]^{2} = (β^{2} - 4 a α) (y^{2} - 4 a x)$

$[∵ T^{2} = S \cdot S_{1}]$

$\Rightarrow β^{2} y^{2} + 4 a^{2} (x^{2} + α^{2} + 2 x \cdot α) - 4 a β y (x + α)$

$= β^{2} y^{2} - 4 β^{2} a x - 4 a α y^{2} + 16 a^{2} α x$

$\Rightarrow β^{2} y^{2} + 4 a^{2} x^{2} + 4 a^{2} α^{2} + 8 x α a^{2}$

$= β^{2} y^{2} - 4 β^{2} a x - 4 a α y^{2} + 16 a^{2} α x - 4 a β x y - 4 a β α y \dots$ (i)

Now, coefficient of $x^{2} = 4 a^{2}$

coefficient of $x y = - 4 a β$

coefficient of $y^{2} = 4 a α$

Again, angle between the two of Eq. (i) is given as $45^{\circ}$

$\begin{array}{lc} ∴ & \tan 45^{\circ} = \frac{2 \sqrt{h^{2} - a b}}{a + b} \\ \Rightarrow & 1 = \frac{2 \sqrt{h^{2} - a b}}{a + b} \\ \Rightarrow & a + b = 2 \sqrt{h^{2} - a b} \\ \Rightarrow & (a + b)^{2} = 4 (h^{2} - a b) \\ \Rightarrow & {(4 a^{2} + 4 a α)}^{2} = 4 [4 a^{2} β^{2} - (4 a^{2}) (4 a α)] \\ \Rightarrow & 16 a^{2} (a + α)^{2} = 4 \cdot 4 a^{2} [β^{2} - 4 a α] \\ \Rightarrow & α^{2} + 6 a α + a^{2} - β^{2} = 0 \\ \Rightarrow & (α + 3 a)^{2} - β^{2} = 8 a^{2} \end{array}$

Thus, the required equation of the locus is $(x + 3 a)^{2} - y^{2} = 8 a^{2}$ which is a hyperbola.

Parabola 3 Question 7

7. The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola, whose

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Answer:

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Parabola 3 Question 7

7. The tangent PT and the normal PN to the parabola y2=4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola, whose

Integer Answer Type Question

Answer:

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Medical Preparation

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Test Platform

Sample Mock Test

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7. The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola, whose