If m is the slope of a common tangent to the curves
\(\frac{x²}{16} + \frac{y²} {9} = 1\)
and x2 + y2 = 12, then 12m2 is equal to:
If m is the slope of a common tangent to the curves
\(\frac{x²}{16} + \frac{y²} {9} = 1\)
and x2 + y2 = 12, then 12m2 is equal to:
6
9
10
12
The Correct Option is B
Solution and Explanation
The correct answer is (B) : 9
\(C1 : \frac{x²}{16} + \frac{y²}{9} = 1\)
and
\(C2 : x² + y² = 12\)
Let
\(y = mx ± \sqrt{16m² + 9}\)
be any tangent to C1 and if this is also tangent to C2 then
\(|\frac{ \sqrt{16m² + 9}}{\sqrt{m² + 1 }}| = √12\)
\(⇒ 16m² + 9 = 12m² + 12\)
\(⇒ 4m² = 3 ⇒ 12m² = 9\)
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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}