Solution : Let | A | = \(\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}\) By using 2×2 determinant formula, we obtain | A | = ( \(sin^2x\)) – (\(-cos^2x\)) = \(sin^2x\) + \(cos^2x\) = 1
Find the determinant of \(\begin{vmatrix} sinx & cosx \\ -cosx & sinx \end{vmatrix}\). Read More »
Walli’s Formula : If m,n \(\in\) N & m, n \(\ge\) 2, then (a) \(\int_{0}^{\pi/2}\) \(sin^nx\)dx = \(\int_{0}^{\pi/2}\) \(cos^nx\)dx = \((n-1)(n-3)….(1 or 2)\over {n(n-2)….(1 or 2)}\) K where K = \(\begin{cases} \pi/2 & \text{if n is even}\ \\ 1 & \text{if n is odd}\ \end{cases}\) (b) \(sin^nx.cos^mx\)dx = \([(n-1)(n-3)….(1 or 2)][(m-1)(m-3)….(1 or 2)]\over {(m+n)(m+n-2)(m+n-4)….(1 or
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Newton Leibnitz formula If h(x) and g(x) are differentiable functions of x then, \(d\over dx\) \(\int_{g(x)}^{h(x)}\) f(t)dt = f[h(x)].h'(x) – f[g(x)].g'(x) Example : Evaluate \(d\over dt\) \(\int_{t^2}^{t^3}\) \(1\over log x\) dx Solution : We have, \(d\over dt\) \(\int_{t^2}^{t^3}\) \(1\over log x\) dx = \(1\over log t^3\) \(d\over dt\) \((t^3)\) – \(1\over log t^2\) \(d\over dt\)
What is Newton Leibnitz formula with Examples ? Read More »
Here you will learn intercept cut by the circle an the axes i.e. x-axis and y-axis respectively. Let’s begin – Intercept Cut by Circle on Axes The intercepts cut by the circle \(x^2 + y^2 + 2gx + 2fy + c\) = 0 on : (i) x-axis x-axis = 2\(\sqrt{g^2 – c}\) (ii) y-axis y-axis
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Here you will learn what is the equation of pair of tangents to a circle from an external point with example. Let’s begin – Equation of Pair of Tangents to a Circle Let the equation of circle S = \(x^2\) + \(y^2\) – \(a^2\) and P(\(x_1,y_1\)) is any point outside the circle. From the point
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Here you will learn what is the length of tangent to a circle formula from an external point with example. Let’s begin – Length of Tangent to a Circle Formula The length of tangent drawn from point (\(x_1,y_1\)) outside the circle S = \(x^2 + y^2 + 2gx + 2fy + c\) = 0 is,
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Parametric Equation of Circle (i) The parametric equation of circle \(x^2 + y^2\) = \(r^2\) are x = rcos\(\theta\), y = rsin\(\theta\) ; \(\theta\) \(\in\) [0,2\(\pi\)) and (rcos\(\theta\), rsin\(\theta\)) are called parametric coordinates. Also Read : General Equation of the Circle – Formula and Examples (ii) The parametric equation of circle \((x – h)^2 +
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Here you will learn what is the equation of director circle of circle with proof. Let’s begin – Director Circle of a Circle The equation of director circle is \(x^2\) + \(y^2\) = 2\(a^2\). Proof : The locus of the point of intersection of two perpendicular tangents to the circle is called director circle. Let
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Solution : Let \(C_1\) be the center of circle \(x^2 + y^2\) = 1 i.e. \(C_1\) = (0, 0) And \(C_2\) be the center of circle \(x^2 + y^2 – 2x – 6y + 6\) = 0 i.e. \(C_2\) = (1, 3) Let \(r_1\) be the radius of first circle and \(r_2\) be the radius
Here you will learn what are orthogonal circles and condition of orthogonal circles. Let’s begin – Orthogonal Circles Let two circles are \(S_1\) = \({x}^2 + {y}^2 + 2{g_1}x + 2{f_1}y + {c_1}\) = 0 and \(S_2\) = \({x}^2 + {y}^2 + 2{g_2}x + 2{f_2}y + {c_2}\) = 0.Then Angle of intersection of two circles
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