The equations of two sides of a variable triangle are \(x-0\) and \(y=3\), and its third side is a tangent to the parabola \(y^2=6 x\). The locus of its circumcentre is:
- $4 y^2-18 y+3 x+18=0$
- $4 y^2-18 y-3 x+18=0$
- $4 y^2+18 y+3 x+18=0$
- $4 y^2-18 y-3 x-18=0$
The Correct Option is C
Approach Solution - 1
To determine the locus of the circumcentre, we consider the properties of the triangle and its relation to the parabola. The triangle’s sides along the y-axis and the line y = 3 form a right angle at the origin. The third side, tangent to the parabola, gives a specific geometric condition that must be analyzed to find the tangent line’s slope and intersection points. Using the derivative of the parabola y2 = 6x:
\(\frac{dy}{dx} = \frac{6}{2y} = \frac{3}{y}\)
Setting this equal to the slope of the tangent line and solving for y and x coordinates of the point of tangency, we can derive the general equation of the tangent line. Subsequent use of circumcentre formulae in a coordinate geometry setting yields the locus as a line equation.
Approach Solution -2
\(y^2=6x\; \&\;y^2=4ax\)
\(⇒4a=6⇒a=23\)
\(y=mx+2m^3;(m\neq0)\)
\(h=\frac{6m−3}{4m^2},k=\frac{6m+3}{4m}\), Now eliminating m and we get
\(⇒3h=2(−2k^2+9k−9)\)
\(⇒4y^2−18y+3x+18=0\)
\(\text{The Correct Option is (C):}\) \(4 y^2+18 y+3 x+18=0\)
Top Questions on Conic sections
- Let \(A\) be the focus of the parabola \(y^2=8x\). Let the line \(y=mx+c\) intersect the parabola at two distinct points \(B\) and \(C\). If the centroid of triangle \(ABC\) is \(\left(\frac{7}{3},\frac{4}{3}\right)\), then \((BC)^2\) is equal to:
- JEE Main - 2026
- Mathematics
- Conic sections
- Let f be a twice differentiable non-negative function such that \((f(x))^2 = 25 + \int_0^x ( f(t)^2 + (f'(t))^2 ) dt\). Then the mean of \(f(\log_2(1)), f(\log_2(2)), \dots, f(\log_2(625))\) is equal to :
- JEE Main - 2026
- Mathematics
- Conic sections
- For some \( \theta\in\left(0,\frac{\pi}{2}\right) \), let the eccentricity and the length of the latus rectum of the hyperbola \[ x^2-y^2\sec^2\theta=8 \] be \( e_1 \) and \( l_1 \), respectively, and let the eccentricity and the length of the latus rectum of the ellipse \[ x^2\sec^2\theta+y^2=6 \] be \( e_2 \) and \( l_2 \), respectively. If \[ e_1^2=\frac{2}{e_2^2}\left(\sec^2\theta+1\right), \] then \[ \left(\frac{l_1l_2}{e_1^2e_2^2}\right)\tan^2\theta \] is equal to:
- JEE Main - 2026
- Mathematics
- Conic sections
- Let \( \vec{c} \) and \( \vec{d} \) be vectors such that \[ |\vec{c} + \vec{d}| = \sqrt{29} \] and \[ \vec{c} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \vec{d}. \] If \( \lambda_1, \lambda_2 \) (\( \lambda_1 > \lambda_2 \)) are the possible values of \[ (\vec{c} + \vec{d}) \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k}), \] then the equation \[ K^2 x^2 + (K^2 - 5K + \lambda_1)xy + \left(3K + \frac{\lambda_2}{2}\right)y^2 - 8x + 12y + \lambda_2 = 0 \] represents a circle, for \( K \) equal to
- JEE Main - 2026
- Mathematics
- Conic sections
- Let \( P(10, 2\sqrt{15}) \) be a point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \] whose foci are \( S \) and \( S' \). If the length of its latus rectum is \(8\), then the square of the area of \( \triangle PSS' \) is equal to
- JEE Main - 2026
- Mathematics
- Conic sections
Questions Asked in JEE Main exam
Which one of the following graphs accurately represents the plot of partial pressure of CS₂ vs its mole fraction in a mixture of acetone and CS₂ at constant temperature?
- JEE Main - 2026
- Organic Chemistry
- For two identical cells each having emf \(E\) and internal resistance \(r\), the current through an external resistor of \(6\,\Omega\) is the same when the cells are connected in series as well as in parallel. The value of the internal resistance \(r\) is ________ \(\Omega\).
- JEE Main - 2026
- Current electricity
- For three unit vectors \( \vec a, \vec b, \vec c \) satisfying \[ |\vec a-\vec b|^2 + |\vec b-\vec c|^2 + |\vec c-\vec a|^2 = 9 \] and \[ |2\vec a + k\vec b + k\vec c| = 3, \] the positive value of \( k \) is:
- JEE Main - 2026
- Vector Algebra
In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$)
- JEE Main - 2026
- Rotational Mechanics
- The system of linear equations
$x + y + z = 6$
$2x + 5y + az = 36$
$x + 2y + 3z = b$
has- JEE Main - 2026
- Matrices and Determinants
Concepts Used:
Sections of a Cone
There are four sections of a cone:
- Circle- It is the locus of a point that moves in a certain plane around a fixed distance. The equation of a circle with center (h,k) and the radius r is: \((x–h)^2 + (y–k)^2 =r^2\)
- Ellipse- It is the set of all points in a plane, the sum of the whose distance from two fixed points in a plane is constant. With the foci on the x-axis, the equation of an ellipse is shown as: \(\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1\)
- Parabola- It is a locus of a point that moves so that its distance from a fixed point is equivalent to the distance from the moving point to fixed straight lines. When the parabola has a focus at (a,0), where, {a > 0} and directrix {x = -a}, its equation is shown as \(y^2 = 4ax\).
- Hyperbola- It is the set of all points in a plane, the difference of whose distance from any two fixed points in the plane is constant. The equation of a hyperbola having its foci on the x-axis is: \(\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\)
Read More: Conic Section