Circles and System of Circles: IITJEE Mains: some solved problems I

Part I: Multiple Choice Questions:

Example 1:

Locus of the mid-points of the chords of the circle x^{2}+y^{2}=4 which subtend a right angle at the centre is (a) x+y=2 (b) x^{2}+y^{2}=1 (c) x^{2}+y^{2}=2 (d) x-y=0

Answer 1: C.

Solution 1:

Let O be the centre of the circle x^{2}+y^{2}=4, and let AB be any chord of this circle, so that \angle AOB=\pi /2. Let M(h,x) be the mid-point of AB. Then, OM is perpendicular to AB. Hence, (AB)^{2}=(OA)^{2}+(AM)^{2}=4-2=2 \Longrightarrow h^{2}+k^{2}=2. Therefore, the locus of (h,k) is x^{2}+y^{2}=2.

Example 2:

If the equation of one tangent to the circle with centre at (2,-1) from the origin is 3x+y=0, then the equation of the other tangent through the origin is (a) 3x-y=0 (b) x+3y=0 (c) x-3y=0 (d) x+2y=0.

Answer 2: C.

Solution 2:

Since 3x+y=0 touches the given circle, its radius equals the length of the perpendicular from the centre (2,-1) to the line 3x+y=0. That is,

r= |\frac{6-1}{\sqrt{9+1}}|=\frac{5}{\sqrt{10}}.

Let y=mx be the equation of the other tangent to the circle from the origin. Then,

|\frac{2m+1}{\sqrt{1+m^{2}}}|=\frac{5}{\sqrt{10}}=25(1+m^{2})=10(2m+1)^{2} \Longrightarrow 3m^{2}+8m-3=0, which gives two values of m and hence, the slopes of two tangents from the origin, with the product of the slopes being -1. Since the slope of the given tangent is -3, that of the required tangent is 1/3, and hence, its equation is x-3y=0.

Example 3.

A variable chord is drawn through the origin to the circle x^{2}+y^{2}-2ax=0. The locus of the centre of the circle drawn on this chord as diameter is (a) x^{2}+y^{2}+ax=0 (b) x^{2}+y^{2}+ay=0 (c) x^{2}+y^{2}-ax=0 (d) x^{2}+ y^{2}-ay=0.

Answer c.

Solution 3:

Let (h,k) be the centre of the required circle. Then, (h,k) being the mid-point of the chord of the given circle, its equation is hx+ky-a(x+h)=h^{2}+k^{2}-2ah.

Since it passes through the origin, we have -ah=h^{2}+k^{2}-2ah \Longrightarrow h^{2}+k^{2}-ah=0.

Hence, locus of (h,k) is x^{2}+y^{2}-ax=0.

Quiz problem:

A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is (a) m(m+n) (b) m+n (c) n(m+n) (d) (1/2)(m+n).

To be continued,

Nalin Pithwa.

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