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  • Prove that the area of the triangle formed by the tangents from the point (h, k) to the parabola x2 = 4ay and a chord of contact isOption: 1

Prove that the area of the triangle formed by the tangents from the point (h, k) to the parabola x2 = 4ay and a chord of contact is

Option: 1

\frac{\left(h^{2}-4 a k\right)^{3 / 2}}{2 a}


Option: 2

\frac{\left(h^{2}-4 a k\right)^{3 / 2}}{4 a}


Option: 3

\frac{\left(h^{2}-4 a k\right)^{3 / 2}}{ a}


Option: 4

\frac{\left(h^{2}-4 a k\right)^{3 / 2}}{3 a}


Answers (1)

best_answer

 

 

Chord of Contact and Diameter of Parabola -

Chord of Contact and Diameter of Parabola

Chord of Contact

S is a parabola and P(x1,y1)  be an external point to parabola S.  A and B are the points of contact of the tangents drawn from P to parabola S. Then the chord AB is called the chord of contact of the parabola S drawn from an external point P.

The equation of the chord of the parabola S=y2-4ax=0 ,  from an external point P(x1,y1) is
\mathbf{T}=\mathbf{0} \text { or } \mathbf{y} \mathbf{y}_{\mathbf{1}}-2 \mathbf{a}\left(\mathbf{x}+\mathbf{x}_{\mathbf{1}}\right)=\mathbf{0}

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Let tangents are drawn from P(h, k) to the parabola x2 = 4ay, intersects the parabola at Q and R.

Then the chord of contact of the tangents to the given parabola is QR.

Then QR is 

x x_{1}=2 a\left(y+y_{1}\right) \\\Rightarrow xh=2a(y+k)\\ \Rightarrow 2ay-xh+2ak=0

Therefore PM = the length of the perpendicular from P(h, k) to QR is

\large \begin{array}{l}{=\left|\frac{2 a k-h^{2}+2 a k}{\sqrt{h^{2}+4 a^{2}}}\right|}\\ \\ {=|-\frac{h^{2}-4 a k}{\sqrt{h^{2}+4 a^{2}} |}} \\\\ {=\left|\frac{h^{2}-4 a k}{\sqrt{h^{2}+4 a^{2}}}\right|}\end{array}

Thus, the area (\DeltaPQR) is 

\begin{array}{l}{=\frac{1}{2} \cdot Q R \cdot P M} \\\\ {=\frac{1}{2} \times \frac{1}{|a|} \sqrt{\left(h^{2}-4 a k\right)\left(h^{2}+4 a^{2}\right)} \times \frac{\left(h^{2}-4 a k\right)}{\sqrt{h^{2}+4 a^{2}}}} \\\\ {=\frac{\left(h^{2}-4 a k\right)^{3 / 2}}{2 a} }\end{array}

 

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