Circles Assignment For JEE, Assignments of Mathematics

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The line ix + my += 0 intersects the curve ax? + 2hxy + by? = 1 at the point P and Q. The circle on PQ as diameter passes through the origin. Prove that #?(a+b)=/? + m?. A(-a.0) ; B(a, 0) are fixed points. C is a point which divides AB in a constant ratiotana. IFAC & CB subtend equal angles at P. prove that the equation of the locus of Pis x7 + y? + 2ax sec2a. + a?=0. A circle is drawn with its centre on the line x + y= 2 to touch the line 4x — 3y + 4 = 0 and pass through the point (0, 1). Find its equation. A point moving around circle (x + 4)? + (y + 2)? = 25 with centre C broke away from it either at the point A or point B on the circle and moved along a tangent to the circle passing through the point D (3, — 3). Find the following. @ Equation of the tangents at A and B. Gi) Coordinates of the points A and B. (ai) Angle ADB and the maximum and minimum distances of the point D from the circle. (i) Area of quadrilateral ADBC and the ADAB. @) Equation of the circle circumscribing the ADAB and also the intercepts made by this circle on the coordinate axes. Find the locus of the mid point of the chord of a circle x? + y?=4 such that the segment intercepted by the chord on the curve x?-2x-—2y=0 subtends a right angle at the origin. Find the equation of a line with gradient 1 such that the two circles x? + y? = 4 and x? +y?— 10x— 14y + 65 = 0 intercept equal length on it. Consider a circle S with centre at the origin and radms 4. Four circles A, B, C and D each with radius unity and centres (-3, 0), (-1, 0), (1, 0) and (3, 0) respectively are drawn. A chord PQ of the circle $ touches the circle B and passes through the centre of the circle C. If the length of this chord can be expressed as ./x . find x. Consider a curve ax? + 2hxy + by? = 1 and a point Pnot on the curve. Aline is drawn from the point P intersects the curve at points Q & R. If the product PQ. PR is independent of the slope of the Ime, then show that the curve is a circle. The line 2x — 3y + 1 = O is tangent to a circle S = 0 at (1, 1). If the radius of the circle is ./73 . Find the equation of the circle S. Find the equation of the circle which passes through the point (1, 1) & which touches the circle xt t+y?+4x-6y-3=0 at the point (2,3) onit. Let a circle be given by 2x(x —a)+y(2y —b) =0, (a#0, b #0). Find the condition on a& b iftwo b chords. each bisected by the x-axis. can be drawn to the circle from the point [a3]. Show that the equation of a straight line meeting the circle x? + y? = a? in two points at equal distances > a, a ‘d’ from a point (x, . y,) on its circumference is xx, + yy) — a? + 77o