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
\(⇒3h=2(−2k^2+9k−9)\)
\(⇒4y^2−18y+3x+18=0\)
\(\text{The Correct Option is (C):}\) \(4 y^2+18 y+3 x+18=0\)
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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: \(\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\)
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