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Locus of Centroid of Tangent Triangle to y²=36x | JEE

JEE Maths question with a full step-by-step solution.

Question
The locus of the centroid of the triangle formed by a tangent to the parabola y2=36xy^2 = 36x with the coordinate axes is:
Ay2=−9xy^2 = -9x
By2+3x=0y^2 + 3x = 0correct
Cy2=3xy^2 = 3x
Dy2=9xy^2 = 9x
Solution
Step 1: Tangent at (x1,y1)(x_1, y_1) is yy1=18(x+x1)yy_1 = 18(x + x_1) (here 2a=182a = 18). Its intercepts are (−x1,0)(-x_1, 0) and (0,18x1y1)\left(0, \dfrac{18x_1}{y_1}\right). Step 2: With the third vertex at OO, the centroid is
x=−x13,y=6x1y1⇒x1=−3x,y1=−18xy.x = -\frac{x_1}{3},\qquad y = \frac{6x_1}{y_1} \Rightarrow x_1 = -3x,\quad y_1 = -\frac{18x}{y}.
 Step 3: Put (x1,y1)(x_1, y_1) back on y12=36x1y_1^2 = 36x_1.
324x2y2=−108x⇒y2=−3x⇒y2+3x=0.\frac{324x^2}{y^2} = -108x \Rightarrow y^2 = -3x \Rightarrow y^2 + 3x = 0.
 Correct answer: (2)
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