Ellipse Questions

What is the parametric equation of ellipse ?

Solution : The equation x = acos\(\theta\) & y = bsin\(\theta\) together represent the parametric equation of ellipse \({x_1}^2\over a^2\) + \({y_1}^2\over b^2\) = 1, where \(\theta\) is a parameter. Note that if P(\(\theta\)) = (acos\(\theta\), bsin\(\theta\)) is on the ellipse then ; Q(\(\theta\)) = (acos\(\theta\), bsin\(\theta\)) is on auxilliary circle. A circle described on …

What is the parametric equation of ellipse ? Read More »

The foci of an ellipse are \((\pm 2, 0)\) and its eccentricity is 1/2, find its equation.

Solution : Let the equation of the ellipse be \(x^2\over a^2\) + \(y^2\over b^2\) = 1. Then, coordinates of the foci are \((\pm ae, 0)\). Therefore,  ae = 2 \(\implies\)  a = 4 We have \(b^2\) = \(a^2(1 – e^2)\) \(\implies\) \(b^2\) =12 Thus, the equation of the ellipse is \(x^2\over 16\) + \(y^2\over 12\) …

The foci of an ellipse are \((\pm 2, 0)\) and its eccentricity is 1/2, find its equation. Read More »

Find the equation of the ellipse whose axes are along the coordinate axes, vertices are \((0, \pm 10)\) and eccentricity e = 4/5.

Solution : Let the equation of the required ellipse be \(x^2\over a^2\) + \(y^2\over b^2\) = 1                 ……….(i) Since the vertices of the ellipse are on y-axis. So, the coordinates of the vertices are \((0, \pm b)\). \(\therefore\)    b = 10 Now, \(a^2\) = \(b^2(1 – e^2)\)  …

Find the equation of the ellipse whose axes are along the coordinate axes, vertices are \((0, \pm 10)\) and eccentricity e = 4/5. Read More »

If the foci of a hyperbola are foci of the ellipse \(x^2\over 25\) + \(y^2\over 9\) = 1. If the eccentricity of the hyperbola be 2, then its equation is :

Solution : For ellipse e = \(4\over 5\), so foci = (\(\pm\)4, 0) for hyperbola e = 2, so a = \(ae\over e\) = \(4\over 2\) = 2, b = \(2\sqrt{4-1}\) = 2\(\sqrt{3}\) Hence the equation of the hyperbola is \(x^2\over 4\) – \(y^2\over 12\) = 1 Similar Questions Find the equation of the ellipse …

If the foci of a hyperbola are foci of the ellipse \(x^2\over 25\) + \(y^2\over 9\) = 1. If the eccentricity of the hyperbola be 2, then its equation is : Read More »

Find the equation of the tangents to the ellipse \(3x^2+4y^2\) = 12 which are perpendicular to the line y + 2x = 4.

Solution : Let m be the slope of the tangent, since the tangent is perpendicular to the line y + 2x = 4 \(\therefore\)  mx – 2 = -1 \(\implies\) m = \(1\over 2\) Since \(3x^2+4y^2\) = 12 or \(x^2\over 4\) + \(y^2\over 3\) = 1 Comparing this with \(x^2\over a^2\) + \(y^2\over b^2\) = …

Find the equation of the tangents to the ellipse \(3x^2+4y^2\) = 12 which are perpendicular to the line y + 2x = 4. Read More »

For what value of k does the line y = x + k touches the ellipse \(9x^2 + 16y^2\) = 144.

Solution : \(\because\) Equation of ellipse is \(9x^2 + 16y^2\) = 144 or \(x^2\over 16\) + \({(y-3)}^2\over 9\) = 1 comparing this with \(x^2\over a^2\) + \(y^2\over b^2\) = 1 then we get \(a^2\) = 16 and \(b^2\) = 9 and comparing the line y = x + k with y = mx + c …

For what value of k does the line y = x + k touches the ellipse \(9x^2 + 16y^2\) = 144. Read More »

Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \(\sqrt{5}\).

Solution : Here S = (2, 3) & S’ is (-2, 3) and b = \(\sqrt{5}\) \(\implies\) SS’ = 4 = 2ae \(\implies\) ae = 2 but \(b^2\) = \(a^2(1-e^2)\) \(\implies\) 5 = \(a^2\) – 4 \(\implies\) a = 3 Hence the equation to major axis is y = 3. Centre of ellipse is midpoint …

Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \(\sqrt{5}\). Read More »