The acute angle between the pair of tangents drawn to the ellipse 2x2 + 3y2 = 5 from the point (1, 3) is
The acute angle between the pair of tangents drawn to the ellipse 2x2 + 3y2 = 5 from the point (1, 3) is
\(\tan^{-1}(\frac{16}{7\sqrt5})\)
\(\tan^{-1}(\frac{24}{7\sqrt{5}})\)
\(\tan^{-1}(\frac{32}{7\sqrt5})\)
\(\tan^{-1}(\frac{3+8\sqrt5}{35})\)
The Correct Option is B
Solution and Explanation
The correct answer is (B) : \(\tan^{-1}(\frac{24}{7\sqrt{5}})\)
2x2 + 3y2 = 5
Equation of tangent having slope m.
\(y=mx±\sqrt{\frac{5}{2}m^2+\frac{5}{3}}\)
which passes through (1, 3)
\(3=m±\sqrt{\frac{5}{2}m^2+\frac{5}{3}}\)
\(\frac{5}{2}m^2+\frac{5}{3}=9+m^2−6m\)
\(\frac{3}{2}m^2+6m−\frac{22}{3}=0\)
\(9m^2+36m−44=0\)
\(m_1+m_2=−4,m_1m_2=−\frac{44}{9}\)
\((m_1−m_2)^2=16+4×\frac{44}{9}=\frac{320}{9}\)
Acute angle between the tangents is given by
\(α=\tan^{−1}|\frac{m_1−m_2}{1+m_1m_2}|\)
\(= \tan^{-1} |\frac{\frac{8\sqrt5}{3}}{1-\frac{44}{9}}|\)
\(= \tan^{-1} (\frac{24\sqrt5}{35})\)
\(α = \tan^{-1}(\frac{24}{7\sqrt5})\)
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Concepts Used:
Ellipse
Ellipse Shape
An ellipse is a locus of a point that moves in such a way that its distance from a fixed point (focus) to its perpendicular distance from a fixed straight line (directrix) is constant. i.e. eccentricity(e) which is less than unity
Properties
- Ellipse has two focal points, also called foci.
- The fixed distance is called a directrix.
- The eccentricity of the ellipse lies between 0 to 1. 0≤e<1
- The total sum of each distance from the locus of an ellipse to the two focal points is constant
- Ellipse has one major axis and one minor axis and a center
Read More: Conic Section
Eccentricity of the Ellipse
The ratio of distances from the center of the ellipse from either focus to the semi-major axis of the ellipse is defined as the eccentricity of the ellipse.
The eccentricity of ellipse, e = c/a
Where c is the focal length and a is length of the semi-major axis.
Since c ≤ a the eccentricity is always greater than 1 in the case of an ellipse.
Also,
c2 = a2 – b2
Therefore, eccentricity becomes:
e = √(a2 – b2)/a
e = √[(a2 – b2)/a2] e = √[1-(b2/a2)]
Area of an ellipse
The area of an ellipse = πab, where a is the semi major axis and b is the semi minor axis.
Position of point related to Ellipse
Let the point p(x1, y1) and ellipse
(x2 / a2) + (y2 / b2) = 1
If [(x12 / a2)+ (y12 / b2) − 1)]
= 0 {on the curve}
<0{inside the curve}
>0 {outside the curve}