Assertion A normal is drawn at a point \) of curve. It meets the -axis and the -axis in point and , respectively, such that where is the origin. The equation of such a curve passing through \) is ^{2}+ \) ^{2}=25 \) Reason and

Both Assertion and Reason are correct and Reason the correct explanation for Assertion

Maths · The solution of differential equation by the method of separation of variables

Assertion A normal is drawn at a point \) of curve. It meets the -axis and the -

Assertion A normal is drawn at a point \( \boldsymbol{P}(\boldsymbol{x}, \boldsymbol{y}) \) of \( \mathbf{a} \) curve. It meets the \( x \) -axis and the \( y \) -axis in point \( A \) and \( B \), respectively, such that \( \frac{1}{O A}+\frac{1}{O B}=1, \) where \( O \) is the origin. The equation of such a curve passing through \( (\mathbf{5}, \mathbf{4}) \) is \( (x-1)^{2}+ \) \( (y-1)^{2}=25 \) Reason \( \boldsymbol{O A}=\boldsymbol{x}+\boldsymbol{y} \frac{\boldsymbol{d} \boldsymbol{y}}{\boldsymbol{d} \boldsymbol{x}} \) and \( \boldsymbol{O} \boldsymbol{B}=\frac{\boldsymbol{x}+\boldsymbol{y} \frac{d \boldsymbol{y}}{d \boldsymbol{x}}}{\frac{d \boldsymbol{y}}{d \boldsymbol{x}}} \)

A. Both Assertion and Reason are correct and Reason the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect

Step-by-step solution

Reason provides the formulas for intercepts OA and OB of the normal: OA = x + y dy/dx and OB = (x + y dy/dx)/(dy/dx). Substituting into the given condition 1/OA + 1/OB = 1 yields the differential equation dy/dx = (x-1)/(1-y). Separating variables and integrating gives (x-1)^2 + (y-1)^2 = constant. Using the point (5,4) gives the constant 25, so the curve is exactly (x-1)^2 + (y-1)^2 = 25. Thus assertion is correct and reason correctly explains it.

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Step-by-step solution

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