Asymptotes of Hyperbola If the length of the perpendicular from a point on a hyperbola to a straight line tends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the asymptote of the hyperbola. The asymptote of a hyperbola can be found as follows. Read more about Asymptotes of Hyperbola[…]
Intersection of a Line and an Ellipse Line: y = mx + c … (1) Ellipse: … (2) Solving equations (1) and (2), we get b²x² + a²(mx + c ) ² = a²b² i.e., (a²m² + b²) x² + 2a² cmx + a²(c² – b²) = 0. Hence, are the tangents to the ellipse Read more about Intersection of a Line and an Ellipse[…]
Equation of Parabola Equation of Parabola when vertex is (h, k) and Axis is Parallel to Y-axis. The equation of the parabola with vertex (h, k) is (x – h)² = 4a (y – k) Its focus is at (h, a + k) and the length of the latus rectum is 4a. The equation of Read more about Equation of Parabola[…]
Equation of an Ellipse Referred to Two Perpendicular Lines Consider the ellipse . Let P (x, y) be any point on the ellipse. Then PM = y and PN = x. . If follows from this if the perpendicular distance P₁ and P₂ of a moving point P (x, y) from two mutually perpendicular coplanar Read more about Equation of an Ellipse Referred to Two Perpendicular Lines[…]
Locus of Point of Intersection of Tangent under Different Conditions Tangent to the parabola y² = 4ax at point P (at₁², 2at₁) and Q (at₂², 2at₂) are t₁ y = x +at₁² and t₂ y = x + at₂² respectively. These tangents intersect at point R (at₁t₂,a (t₁ + t₂). If we want to find Read more about Locus of Point of Intersection of Tangent under Different Conditions[…]
Segments of Secants, Chord and Tangent Segments of Secants and Chord: Secants AB and CD intersect inside the circle in fig (a) and outside the circle in fig (b). From the figure, we have ∠DCB = ∠DAB and ∠ADC = ∠ABC Hence ΔADE ~ ΔCBE ∴ . AE x BE = CE x DE Segments Read more about Segments of Secants, Chord and Tangent[…]
Concurrency of Three Lines Three linesare said to be concurrent if they pass through a common point, i.e. they meetat a point. Thus, if three lines are concurrent the point of intersection of the two lines on the third line. Let the three concurrent lines be a₁x + b₁y + c₁ = 0 a₂x + Read more about Concurrency of Three Lines[…]
Common Chord of Two Circles The common chord joining the point of intersection of the two given circles is called their common chord. If S = 0 and S’ = 0 are two intersecting circles, the equation of their common chord is S – S’ = 0 Let S = x² + y² + 2gx Read more about Common Chord of Two Circles[…]
Position of a Point with respect to a Circle Let thecircle be x²+y² + 2gx +2fy + c = 0 Point P lies outside on, or inside thecircle according to as CP is greater than, equal to or less than radius,respectively I.e. . S₁ = x₁² + y₁² +2gx₁ + 2fy₁ + c = >, Read more about Position of a Point with respect to a Circle[…]
Avalanche Breakdown If the reverse – bias applied to a p – n junction is increased, a point will reach when the junction breaks down and reverse current rises sharply to a value limited only by the external resistance connected in series. This specific value of the reverse bias voltage is called breakdown voltage (V). Read more about Avalanche Breakdown[…]