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Slope Form of Tangent: Hyperbola

The area of t...

Question

The area of triangle formed by the tangents at the points $t_{1}, t_{2}, t_{3}$ on the parabola $y^{2} = 4 a x$

a^{2} | (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) |

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2 a^{2} | (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) |

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\frac{a^{2}}{2} | (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) |

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a^{2} | t_{1} t_{2} t_{3} |

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Solution

The correct option is C $\frac{a^{2}}{2} | (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) |$
Three points are: $(a t_{1}^{2}, 2 a t_{1}), (a t_{2}^{2}, 2 a t_{2}), (a t_{3}^{2}, 2 a t_{3})$
$A = \frac{1}{2} ∣ ∣ ∣ ∣ ∣ \begin{matrix} a {t_{1}}^{2} & 2 a t_{1} & 1 a {t_{2}}^{2} & 2 a t_{2} & 1 a {t_{3}}^{2} & 2 a t_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = \frac{a^{2}}{2} | (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) |$

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The area of triangle formed by the tangents at the points t1,t2,t3 on the parabola y2=4ax

The correct option is C a22|(t1−t2)(t2−t3)(t3−t1)|Three points are: (at21,2at1),(at22,2at2),(at23,2at3)A=12∣∣ ∣ ∣∣at122at11at222at21at322at31∣∣ ∣ ∣∣=a22|(t1−t2)(t2−t3)(t3−t1)|

The area of triangle formed by the tangents at the points $t_{1}, t_{2}, t_{3}$ on the parabola $y^{2} = 4 a x$