Q57 / 150 · MHT CET 2026 (16 April Shift 1) · Mathematics
The equations of the tangent to the curve $x^2 + y^2 = 10$, where the tangent is parallel to the line $2x + y - 1 = 0$, are
- A$2x + y = \pm\sqrt{2}$
- Correct answerB$2x + y = \pm 5\sqrt{2}$
- C$2x + y = \pm 2\sqrt{2}$
- D$2x + y = \pm 3\sqrt{2}$
Solution
The given curve is a circle $x^2 + y^2 = 10$ with center $(0,0)$ and radius $r = \sqrt{10}$.
The tangent is parallel to the line $2x + y - 1 = 0$. Let the equation of the tangent be $2x + y + c = 0$.
The perpendicular distance from the center of the circle to the tangent is equal to its radius.
$\dfrac{|2(0) + (0) + c|}{\sqrt{2^2 + 1^2}} = \sqrt{10}$
$\dfrac{|c|}{\sqrt{5}} = \sqrt{10}$
$|c| = \sqrt{50} = 5\sqrt{2}$
$c = \pm 5\sqrt{2}$
The equations of the tangents are $2x + y = \pm 5\sqrt{2}$.
Answer: $2x + y = \pm 5\sqrt{2}$
The tangent is parallel to the line $2x + y - 1 = 0$. Let the equation of the tangent be $2x + y + c = 0$.
The perpendicular distance from the center of the circle to the tangent is equal to its radius.
$\dfrac{|2(0) + (0) + c|}{\sqrt{2^2 + 1^2}} = \sqrt{10}$
$\dfrac{|c|}{\sqrt{5}} = \sqrt{10}$
$|c| = \sqrt{50} = 5\sqrt{2}$
$c = \pm 5\sqrt{2}$
The equations of the tangents are $2x + y = \pm 5\sqrt{2}$.
Answer: $2x + y = \pm 5\sqrt{2}$
Hence, the correct answer is option: 'B'.