Let P(a, b) be a point on the parabola y2 = 8x such that the tangent at P passes through the centre of the circle x2 + y2 – 10x – 14y + 65 = 0. Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to
- 0
- 25
- 40
- 65
The Correct Option is D
Solution and Explanation
The correct answer is (D):
Centre of circle x2 + y2 – 10x –14y + 65 = 0 is at (5, 7).
Let the equation of tangent to y2 = 8x is
yt = x + 2t2
which passes through (5, 7)
7t = 5 + 2t2
⇒ 2t2 – 7t + 5 = 0
t = 1, \(\frac{5}{2}\)
A = 1×12×2×(\(\frac{5}{2}\))2
= 25
B = 2×2×1×2×2×\(\frac{5}{2}\)
= 40
A+B = 65
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Concepts Used:
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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.
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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.
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