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At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).
  • a)
    (x + 3)2 = y + 4
  • b)
    (x + 5)2 = 2y + 3
  • c)
    (x + 4)2 = y + 3
  • d)
    (x + 5)2 = 2y + 3
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
At any point (x, y) of a curve, the slope of the tangent is twice the ...
Slope of the line segment joining the point of contact P (x , y) to the point (- 4 , - 3) = 



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At any point (x, y) of a curve, the slope of the tangent is twice the ...
Understanding the Problem
- We need to find a curve where the slope of the tangent at any point (x, y) is twice the slope of the line segment connecting (x, y) to the fixed point (4, 3).
- The curve also passes through the point (2, 1).
Finding the Slopes
- The slope of the tangent at point (x, y) is represented as dy/dx.
- The slope of the line segment from (x, y) to (4, 3) is calculated as (3 - y) / (4 - x).
Setting Up the Equation
- According to the problem, we have:
dy/dx = 2 * ((3 - y) / (4 - x))
Rearranging the Equation
- Rearranging the equation gives:
dy/dx = (2(3 - y)) / (4 - x)
Separation of Variables
- To solve this differential equation, we can rearrange the terms:
(4 - x) dy = 2(3 - y) dx
Integrating Both Sides
- Integrate both sides to find the equation of the curve:
∫(4 - x) dy = ∫2(3 - y) dx
- This results in:
y(4-x) = 6x - x^2 + C
Using Initial Condition
- With the condition that the curve passes through (2, 1), substitute these values to find C:
1(4-2) = 6(2) - (2^2) + C
2 = 12 - 4 + C
C = -6
Final Equation of the Curve
- Substitute C back into the equation:
y(4 - x) = 6x - x^2 - 6
- Rearranging gives the form:
(x + 4)² = y + 3
- Thus, the equation of the curve is:
(x + 4)² = y + 3, which corresponds to option C.
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