Examples On Tangents To Circles Set-5

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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 \(\begin{align}\frac{{2\lambda k}}{{{k^2} - {g^2}}}\end{align}\).