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Problem 18

The circles \(x^{2}+y^{2}+2 a x-c^{2}=0\) and \(x^{2}+y^{2}+2 b x-c^{2}=0\) intersect at \(A\) and B. A line through \(A\) meets one circle at \(P\) and a parallel line through \(B\) meets the other circle at Q. Show that the locus of the mid point of \(P Q\) is a circle.

Expert verified

Hence, it has been proven that the locus of the midpoint of \(PQ\) is a circle.

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Chapter 1

Find the locus of the point of intersection of two perpendicular lines each of which touches one of the two circles \((x-a)^{2}+y^{2}=b^{2},(x+a)^{2}+y^{2}=c^{2}\) and prove that the bisectors of the angles between the straight lines always touch one or the other fixed circles.

Chapter 1

The points \(A(x, y), B(y, z)\) and \(C(x, x)\) represents the vertices of a right angled triangle, if : (a) \(x=y\) (b) \(y=z\) (c) \(\mathrm{z}=x\) (d) \(x=y=z\)

Chapter 1

Let \(A\) be the centre of the circle \(x^{2}+y^{2}-2 x-4 y-20=0\). Suppose that the tangents at the points \(B(1,7)\) and \(D(4,-2)\) on the circle meet at the point \(C\). Find the area of the quadrilateral \(A B C D\).

Chapter 1

If the point $\left[x_{1}+t\left(x_{2}-x_{1}\right),
y_{1}+t\left(y_{2}-y_{1}\right)\right]\( divides the join of \)\left(x_{1},
y_{1}\right)\( and \)\left(x_{2}, y_{2}\right)$ internally, then:
(b) \(0

Chapter 1

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