Question:
Let the tangents at the points P and Q on the ellipse\(\begin{array}{l} \frac{x^2}{2}+\frac{y^2}{4}=1\ \text{meet at the point}\ R\left(\sqrt{2},2\sqrt{2}-2\right).\end{array}\)If S is the focus of the ellipse on its negative major axis, then SP2 + SQ2 is equal to ________.
Let the tangents at the points P and Q on the ellipse
\(\begin{array}{l} \frac{x^2}{2}+\frac{y^2}{4}=1\ \text{meet at the point}\ R\left(\sqrt{2},2\sqrt{2}-2\right).\end{array}\)
If S is the focus of the ellipse on its negative major axis, then SP2 + SQ2 is equal to ________.
Updated On: Feb 7, 2025
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Solution and Explanation
\(E≡\frac{x^2}{2}+\frac{y^2}{4}=1\)
\(\begin{array}{l} T\equiv y=mx\pm \sqrt{2m^2+4}\end{array}\)passes through \((\sqrt2,2\sqrt2−2)\)
\(\begin{array}{l} T\equiv y=mx\pm \sqrt{2m^2+4}\end{array}\)passes through \((\sqrt2,2\sqrt2−2)\)
\(\begin{array}{l} \Rightarrow\ \left(2\sqrt{2}-2-m\sqrt{2}\right)=\pm\sqrt{2m^2+4} \end{array}\)
\(\begin{array}{l} \Rightarrow 2m^2-2m\sqrt{2}\left(2\sqrt{2}-2\right)+4\left(3-2\sqrt{2}\right)=2m^2+4\end{array}\)
\(\begin{array}{l} \Rightarrow\ -2\sqrt{2}m\left(2\sqrt{2}-2\right)=4-12+8\sqrt{2}\end{array}\)
\(\begin{array}{l} \Rightarrow\ -4\sqrt{2}m\left(\sqrt{2}-1\right)=8\left(\sqrt{2}-1\right)\end{array}\)
\(\begin{array}{l} \Rightarrow\ m=-\sqrt{2}\text{ and }m\rightarrow\infty\end{array}\)
\(\begin{array}{l}\therefore\text{Tangents are}~ x=\sqrt{2}\text{ and }y=-\sqrt{2}x+\sqrt{8} \end{array}\)
\(\begin{array}{l} \therefore\ P\left(\sqrt{2},0\right)\text{ and }Q\left(1,\sqrt{2}\right)\end{array}\)and \(\begin{array}{l} S=\left(0,-\sqrt{2}\right)\end{array}\)
\(\begin{array}{l}\therefore \left(PS\right)^2 + \left(QS\right)^2 = 4 + 9 = 13\end{array}\)
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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}