If four distinct points (2, 3), (0, 2), (4, 5) and (0, t) are concylic, then t3+ 17 is equal to
Let equation of the circle be x2 + y2 + 2gx + 2fy + c = 0. Since it passes through the points (2, 3), (0, 2), (4, 5) we have
4g + 6f + c = –13, 4f + c = –4, 8g + 10f + c = –41. Solving these equations we get g = 5/2, f = –19/2, c = 34 = 0. As the circle passes through the point (0, t), t2 – 19t + 34 = 0 ⇒ t = 2 or t = 17. But t = 2 corresponds to the point (0, 2) which different from (0, t). Therefore t = 17 and t3 + 17 = 4930.
A chord of the circle x2 + y2 – 4x – 6y = 0 passing through the origin subtends an angle tan-1 (7/4) at the point where the circle meets positivey-axis.
Equation of the chord is
On the line joining the points A (0, 4) and B (3, 0), a square ABCD is constructed on the side of the line away from the origin. Equation of the circle having centre at C and touching the axis of x is
A circle with centre at the origin and radius equal to a meets the axis of x at A and B.P (α) and Q (β) are two points on this circle so that α – β = 2γ, where γ is a constant. The locus of the point of intersection of APand BQ is
A circle C1 of radius b touches the circle x2 + y2 = a2 externally and has its centre on the positive x-axis; another circle C2 of radius c touches the circle C1 externally and has its centre on the positive x-axis. Given a< b < c, then the three circles have a common tangent if a, b, c are in
Angle of intersection of these circle is
If C1, C2 are the centre of these circles than area of Δ OC1 C2, where Ois the origin, is
A triangle has two of its sides along the axes, its third side touches the circle x2 + y2 – 2ax – 2ay + a2 = 0. If he locus of the circumcentre of the triangle passes through the point (38, –37) then a2 – 2a is equal to
Two circles are inscribed and circumscribes about a square ABCD, length of each side of the square is 32. P and Q are points respectively of these circles, then is equal to
Let A be the centre of the circle x2 + y2 – 2x – 4y – 20 = 0. The tangents at the point B(1, 7) and C(4, –2) on the circle meet at the point D. If Δ denotes the area of the quadrilateral ABCD, then 45Δ is equal to
Equation of a tangent to the circle with centre (2, –1) is 3x + y = 0. The squar of the length of the tangent to the circle from the point (23, 17) is