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Chapter 1: Problem 21
If the distance of any point \((x, y)\) from the origin is defined as \(d(x, y)=\max \\{|x|,|y|\\}\) \(d(x, y)=\) a non-zero constant, then the locus is : (a) a circle (b) a straight line (c) a square (d) a triangle
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If \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y, f(1)=2\) and \(\alpha_{n}=f(n), n \in N\) then the equation of the circle having \(\left(\alpha_{1}, \alpha_{2}\right)\) and \(\left(\alpha_{3}, \alpha_{4}\right)\) as the ends of its one diameter is : (a) \((x-2)(x-8)+(y-4)(y-16)=0\) (b) \((x-4)(x-8)+(y-2)(y-16)=0\) (c) \((x-2)(x-16)+(y-4)(y-8)=0\) (d) \((x-6)(x-8)+(y-5)(y-6)=0\)
Find the equation of the pair of tangents from the origin to the circle \(x^{2}+y^{2}+2 g x+2 f y+\lambda^{2}=0\), and show that their intercept on the line \(y=h\) iss \(\frac{2 h \lambda}{\lambda^{2}-g^{2}}\) times the radius of the circle.
If each of the vertices of a triangle has integral co-ordinates then the triangle may be: (a) right angled (b) equilateral (c) isosceles (d) scalene
An isosceles right angled triangle, whose sides are \(1,1, \sqrt{2}\) lies entirely in the first quadrant with the ends of the hypotenuse on the co- ordinate axes. If it slides, prove that the locus of its centroid is $$ (3 x-y)^{2}+(x-3 y)^{2}=\frac{32}{9} $$
The equation of the circle which touches the axes of co-ordinates and the line \(\frac{x}{3}+\frac{y}{4}=1\) and whose centres lies in the first quadrant is $$ x^{2}+y^{2}-2 \lambda x-2 \lambda y+\lambda^{2}=0 $$ which \(\lambda\) is equal to: (a) 1 (b) 2 (c) 3 (d) 6
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