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
Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.
- JEE Main - 2026
- Coordinate Geometry
- A voltage regulating circuit consisting of a Zener diode having breakdown voltage of $10\,\text{V}$ and maximum power dissipation of $0.4\,\text{W}$ is operated at $15\,\text{V}$. The approximate value of protective resistance in this circuit is ___ $\Omega$.
- JEE Main - 2026
- Semiconductor electronics: materials, devices and simple circuits
- Two point charges of \(1\,\text{nC}\) and \(2\,\text{nC}\) are placed at two corners of an equilateral triangle of side \(3\) cm. The work done in bringing a charge of \(3\,\text{nC}\) from infinity to the third corner of the triangle is ________ \(\mu\text{J}\). \[ \left(\frac{1}{4\pi\varepsilon_0}=9\times10^9\,\text{N m}^2\text{C}^{-2}\right) \]
- JEE Main - 2026
- Electrostatics
Consider the following two reactions A and B:
The numerical value of [molar mass of $x$ + molar mass of $y$] is ___.
- JEE Main - 2026
- Organic Chemistry
Consider an A.P. $a_1,a_2,\ldots,a_n$; $a_1>0$. If $a_2-a_1=-\dfrac{3}{4}$, $a_n=\dfrac{1}{4}a_1$, and \[ \sum_{i=1}^{n} a_i=\frac{525}{2}, \] then $\sum_{i=1}^{17} a_i$ is equal to
- JEE Main - 2026
- Arithmetic Progression
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