Question:
If the lines joining the origin to the points of intersection of the line $y = mx + 1$ and the circle $x^2 + y^2 = 1$ are perpendicular to each other, then the value of $m^2$ is:
If the lines joining the origin to the points of intersection of the line $y = mx + 1$ and the circle $x^2 + y^2 = 1$ are perpendicular to each other, then the value of $m^2$ is:
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Homogenization reduces the perpendicularity condition of the intersecting lines to simply setting the sum of the $x^2$ and $y^2$ coefficients to zero: $(1-m^2) + 0 = 0 \implies m^2 = 1$.
Updated On: Jun 3, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Concept
To find the equation of the lines joining the origin to the intersection points of a curve and a line, we homogenize the equation of the curve to degree 2 using the equation of the line.
Step 2: Meaning
The line equation is $y - mx = 1$. We use this relationship to homogenize the circle equation $x^2 + y^2 = 1$.
Step 3: Analysis
Homogenizing the circle equation: \[ x^2 + y^2 = 1^2 \implies x^2 + y^2 = (y - mx)^2 \] Expanding the right-hand side: \[ x^2 + y^2 = y^2 - 2mxy + m^2x^2 \] \[ x^2(1 - m^2) + 2mxy = 0 \] Since the lines are perpendicular to each other, the sum of the coefficients of $x^2$ and $y^2$ must equal zero: \[ \text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0 \] \[ (1 - m^2) + 0 = 0 \implies m^2 = 1 \]
Step 4: Conclusion
The value of $m^2$ is $1$.
Final Answer: (A)
To find the equation of the lines joining the origin to the intersection points of a curve and a line, we homogenize the equation of the curve to degree 2 using the equation of the line.
Step 2: Meaning
The line equation is $y - mx = 1$. We use this relationship to homogenize the circle equation $x^2 + y^2 = 1$.
Step 3: Analysis
Homogenizing the circle equation: \[ x^2 + y^2 = 1^2 \implies x^2 + y^2 = (y - mx)^2 \] Expanding the right-hand side: \[ x^2 + y^2 = y^2 - 2mxy + m^2x^2 \] \[ x^2(1 - m^2) + 2mxy = 0 \] Since the lines are perpendicular to each other, the sum of the coefficients of $x^2$ and $y^2$ must equal zero: \[ \text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0 \] \[ (1 - m^2) + 0 = 0 \implies m^2 = 1 \]
Step 4: Conclusion
The value of $m^2$ is $1$.
Final Answer: (A)
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