A common tangent T to the curves
\(C_1:\frac{x^2}{4}+\frac{y^2}{9} = 1\)
and
\(C_2:\frac{x^2}{4^2}\frac{-y^2}{143} = 1\)
does not pass through the fourth quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to ______.
A common tangent T to the curves
\(C_1:\frac{x^2}{4}+\frac{y^2}{9} = 1\)
and
\(C_2:\frac{x^2}{4^2}\frac{-y^2}{143} = 1\)
does not pass through the fourth quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to ______.
Correct Answer: 20
Solution and Explanation
Equation of tangent to ellipse
\(\frac{x^2}{4}+\frac{y^2}{9} = 1\)
and given slope m is :
\(y = mx + \sqrt{4m^2+9}...(i)\)
For slope m equation of tangent to hyperbola is :
\(y = mx+\sqrt{42m^2-143}...(ii)\)
Tangents from (i) and (ii) are identical then
4m2 + 9 = 42m2 – 143
∴ m = ±2 (+2 is not applicable)
∴ m = -2
Hence
x1 = \(\frac{8}{5}\)
and
x2 = \(\frac{84}{5}\)
\(∴ |2x_1+x_2| = |\frac{16}{5}+\frac{84}{5}|\)
= 20
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Concepts Used:
Types of Differential Equations
There are various types of Differential Equation, such as:
Ordinary Differential Equations:
Ordinary Differential Equations is an equation that indicates the relation of having one independent variable x, and one dependent variable y, along with some of its other derivatives.
\(F(\frac{dy}{dt},y,t) = 0\)
Partial Differential Equations:
A partial differential equation is a type, in which the equation carries many unknown variables with their partial derivatives.
Linear Differential Equations:
It is the linear polynomial equation in which derivatives of different variables exist. Linear Partial Differential Equation derivatives are partial and function is dependent on the variable.
Homogeneous Differential Equations:
When the degree of f(x,y) and g(x,y) is the same, it is known to be a homogeneous differential equation.
\(\frac{dy}{dx} = \frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\)
Read More: Differential Equations