Name: Pair of Tangent, Chord of Contact
Uploaded: 2019-07-19
Duration: 19 min 37 s
Description: This video contains all the details of the chord of the contact pair of tangent and cord bisected at a point. This concept is very useful for the students.

Question

One vertex of a triangle of given species is fixed, and another moves along the circumference of a fixed circle; prove that the locus of the remaining vertex is a circle, and find its radius.

Accepted Answer

Let $OPQ$ be any triangle, where $O$ is the origin. $Q$ is any point on the circumference of a circle whose centre is $C$ $(a, 0)$ , and passes through origin.

The polar coordinates of $C$ will be $(a, 0)$ , and for $Q (r_{1}, α_{1}) .$

Assume that $P$ be the moving point $(r, α) .$

Let the angles $O P Q,$ $O Q P$ and $P O Q$ be $θ_{1}, θ_{2}$ and $θ_{3}$ respectively of the $∆ O P Q$ .

Now in triangle $∆ O C Q,$ $\cos α_{1} = \frac{O C^{2} + O Q^{2} - C Q^{2}}{2 O C . O Q}$

$\Rightarrow$ $\cos α_{1} = \frac{a^{2} + {r_{1}}^{2} - d^{2}}{2 a . r_{1}}$ (Say $C Q = d$ )

$\Rightarrow$ $d^{2} = {r_{1}}^{2} + a^{2} - 2 r_{1} a \cos α_{1} \dots (1)$

Now we have to eliminate $r_{1}$ and $α_{1}$ from $(1)$ , we get:

$∠ C O Q = ∠ P O C - ∠ P O Q$

$\Rightarrow$ $α_{1} = (α - θ_{3})$ $\dots (2)$

Again, from $∆ O P Q,$ by Sine Rule, we get:
$\frac{O P}{\sin θ_{2}} = \frac{O Q}{\sin θ_{3}}$

$\Rightarrow$ $\frac{r_{}}{\sin θ_{2}} = \frac{r_{1}}{\sin θ_{3}}$

$\Rightarrow$ $r_{1} = \frac{r \sin θ_{3}}{\sin θ_{2}}$ $\dots (3)$

Putting the values of $α_{1}$ and $r_{1}$ from $(2)$ and $(3)$ in $(1)$ , we get:

$d^{2} = {\{\frac{r \sin θ_{3}}{\sin θ_{2}}\}}^{2} + a^{2} - 2 \frac{r \sin θ_{3_{}}}{\sin θ_{2}} a \cos (α - θ_{3})$

$\Rightarrow$ $\frac{d^{2} \sin^{2} θ_{2}}{\sin^{2} θ_{3}} = r^{2} + a^{2} \frac{\sin^{2} θ_{2}}{\sin^{2} θ_{3}} - 2 a r \frac{\sin θ_{2}}{\sin θ_{3}} \cos (α - θ_{3})$ .

The equation represents a circle with a radius $\frac{\sin θ_{2}}{\sin θ_{3_{}}}$ .

One vertex of a triangle of given species is fixed, and another moves along the circumference of a fixed circle; prove that the locus of the remaining vertex is a circle, and find its radius.

Important Questions on The Circle

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