# Examples On Tangents To Circles Set-5

**Example - 29**

Let \({C_1}\) and \({C_2}\) be two circles with \({C_2}\) lying inside \({C_1}.\) *A* circle \({C}\) lying inside \({C_1}\) touches \({C_1}\) internally and \({C_2}\) externally. Determine the locus of the centre of \({C}\).

**Solution:** First of all, you must note that is not concentric with \({C_1}.\) All that is said is that \({C_2}\) lies *somewhere *inside \({C_1}.\)

Let us first discuss what all would be involved in solving this question through a co-ordinate approach.

We could first assume a co-ordinate axes, say, with the *\(x\)*-axis lying along the line joining the centres of \({C_1}\)\({C_1}\) and \({C_2}\) and the origin at the centre of \({C_1}:\)

We could then assume the equation of \(C \)to be \({x^2} + {y^2} + 2gx + 2fy + c = 0\)and impose the necessary constraints on *\(g\)*, *\(f\)* and *\(c\) *so that* \(C\)* touches \({C_2}\) externally and \({C_1}\) internally. Recall the appropriate constraints for two circles \({S_1}\) and \({S_2}\) touching externally and internally.

Imposing these constraints gives us the necessary conditions that *\(g\)*, *\(f\)** *and *\(c\)* must satisfy and hence the locus of the centre of *\(C\)* which is \(( - g, - f).\)

However, we will be much letter off in using a pure-geometry approach here also as you’ll soon see.

Assume an arbitrary circle *\(C\)* of radius *\(r\)* inside \({C_1}\) with centre *\(X\)*, and which satisfies the given constraint:

By the properties of circles touching internally and externally, we have

\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&{O_1}X = {O_1}A - AX = {r_1} - r\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&{O_2}X = {O_2}B + BX = {r_2} + r\\ \Rightarrow \qquad & {O_1}X + {O_2}X = {r_1} + {r_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)\end{array}

(1) simply states the centre of *\(C\)* i.e. * \(x\)*, moves in such a way so that the sum of its distances from \({O_1}\) and \({O_2}\) is constant. Thus, *\(X\)* must lie on an ellipse with \({O_1}\) and \({O_2}\) as the two foci !

To sketch the path that *\(X\)* can take, we can follow the approach described in the unit on Complex numbers. Fix two pegs at \({O_1}\) and \({O_2}\) and tie a string of length \({r_1} + {r_2}\) between these two pegs. Use your pencil and the taut string as a guide to trace out the ellipse. This is the path on which the centre of *\(C\)* can move.

**Example - 30 **

**POLE AND POLAR**

Through an arbitrary fixed point \(P({x_1},{y_1}),\) a variable line is drawn intersecting the circle \({x^2} + {y^2} = {a^2}\) at *\(A\)* and *\(B\)* respectively. Tangents drawn to this circle at *\(A\)* and *\(B\)* intersect at *\(Q\)*. Find the locus of *\(Q\)*.

**Solution:**

We can view this situation from a different perspective. We have a point \(Q(h,k)\) from which we draw tangents *\(QA\) *and *\(QB\) *to the circle \({x^2} + {y^2} = {a^2}.\) Thus, the equation of *\(AB\) *(which is the chord of contact) is

\[\begin{align}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\&T(h,k) = 0 \\ \Rightarrow & hx + ky = {a^2}\end{align}\]

Now, this chord of contact also passes through \(P({x_1},{y_1})\) so that

\[h{x_1} + k{y_1} = {a^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(1)\]

What we have in (1) is a linear equation involving the variables *\(h\)* and *\(k\)*. Note that and are constant. Thus, we can infer from (1) that \((h,k)\) lies on the line

\[x{x_1} + y{y_1} = {a^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(2)\]

This is the required equation !

The line obtained in (2) is referred to as the **polar** of the point *\(P\)* with respect to the given circle. *\(P\)* is itself referred to as the **pole** of the polar.

Notice that the equation of the polar can be written concisely as

\(\qquad \qquad \fbox{$\begin{array}{*{20}{c}} {T({x_1},{y_1}) = 0} \end{array}$}\):Equation of the polar for the pole P\(\left( {{x_1},{y_2}} \right)\)

This example should show you that sometimes enormous simplifications are achieved using a co-ordinate geometrical approach rather than a pure-geometrical one. Co-ordinate geometry is not all that bad!

# TRY YOURSELF - I

**Q. 1** Three circle with radii \({r_1},{r_2}\) and \({r_3}\) touch each other externally. The tangents at their points of contact meet at a point whose distance from any point of contact is \(4\). Show geometrically that

\[\frac{{{r_1}{r_2}{r_3}}}{{{r_1} + {r_2} + {r_3}}} = 16\]

**Q. 2** What is the equation of the tangent to \({x^2} + {y^2} - 30x + 6y + 109 = 0\) at \(({4_1} - 1)?\)

**Q. 3** Find the tangents to \({x^2} + {y^2} - 6x + 4y - 12 = 0\) parallel to \(4x + 3y + 5 = 0.\)

**Q. 4** Find the tangents to \({x^2} + {y^2} = {a^2}\) which make a triangle of area \({a^2}\) with the axes.

**Q. 5** If \(P(a,b)\) and \(Q(b,a)\) are two points \((b \ne a),\) find the equation of the circle touching *\(OP\)* and *\(OQ\) *at *\(P\)* and *\(Q\)* where *\(O\)* is the origin .

**Q. 6** Find the equation of the normal to \(3{x^2} + 3{y^2} - 4x - 6y = 0\) at (0, 0)

**Q. 7** The line \(2x - y + 1 = 0\) is a tangent to a circle at \((2, 5)\); its centre lies on \(x + y = 9.\) Find its equation.

**Q. 8** If \(3x + y = 0\) is a tangent to a circle whose centre is \((2, –1)\), find the other tangent to the circle from the origin.

**Q. 9** Find the locus of the point of intersection of tangents to the circle \({x^2} + {y^2} + 4x - 6x + 9{\sin ^2}\alpha + 13{\cos ^2}\alpha = 0\) which are inclined at an angle of \(2\alpha \) to each other.

**Q. 10** Prove that the intercept of the pair of tangents from the origin to \({x^2} + {y^2} + 2gx + 2fy + {k^2} = 0,\) on the line \(y = \lambda \) is \(\begin{align}\frac{{2\lambda k}}{{{k^2} - {g^2}}}\end{align}\).