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Problem 3

The sides of a triangle are \(3 x+4 y, 4 x+3 y\) and \(5 x+5 y\) units, where \(x>0, y>0\). The triangle is: (a) right angled (b) acute angled (c) obtuse angled (d) isosceles

Expert verified

The triangle is a right angled triangle, but not an isosceles triangle.

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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 circle \(x^{2}+y^{2}+2 g x+2 f y+c=0\) cuts each of the circles \(x^{2}+y^{2}-4=0\), \(x^{2}+y^{2}-6 x-8 y+10=0\) and \(x^{2}+y^{2}+2 x-4 y-2=0\) at the extremities of a diameter, then : (b) \(g+f=c-1\) (a) \(c=-4\) (d) \(g f=6\) (c) \(g^{2}+f^{2}-c=17\)

Chapter 1

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) \((x-6)(x-8)+(y-5)(y-6)=0\)

Chapter 1

The equation of tangents drawn from the origin to the circle $x^{2}+y^{2}-2 r x-2 h y+h^{2}=0$, are : (a) \(x=0\) (b) \(y=0\) (c) \(\left(h^{2}-r^{2}\right) x-2 r h y=0\) (d) \(\left(h^{2}-r^{2}\right) x+2 r h y=0\)

Chapter 1

A rectangle \(A B C D\) is inscribed in a circle with a diameter lying along the line \(3 y=x+10\). If \(A\) and \(B\) are the points \((-6,7)\) and \((4,7)\) respectively, find the area of the rectangle.

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