Qus : 1 2
A point P in the first quadrant, lies on $y^2 = 4ax$, a > 0, and keeps a distance of 5a units from its focus. Which of the following points lies on the locus of P?
1 (1,0) 2 (1,1) 3 (0,2) 4 (2,0) Go to Discussion Solution
Locus of Point on Parabola
Given: Point on parabola \( y^2 = 4a x \) is at distance \( 5a \) from focus \( (a, 0) \).
Distance Equation:
\[
(x - a)^2 + y^2 = 25a^2
\Rightarrow (x - a)^2 + 4a x = 25a^2
\Rightarrow x^2 + 2a x - 24a^2 = 0
\]
Solving gives: \( x = 4a \), \( y = 4a \)
✅ Final Answer:
\( \boxed{(4a,\ 4a)} \)
Qus : 2 3 A circle touches the x–axis and also touches the circle with centre (0, 3) and radius 2. The locus of the centre of the circle is
1 a circle 2 an ellipse 3 a parabola 4 a hyperbola Go to Discussion
Solution Qus : 3 2 The two parabolas $y^2 = 4a(x + c)$ and $y^2 = 4bx, a > b > 0$ cannot
have a common normal unless
1 $$c>2(a+b)$$ 2 $$c>(a-b)$$ 3 $$c<2(a-b)$$ 4 $$c<\frac{2}{a-b}$$ Go to Discussion
Solution Qus : 4 2 Coordinate of the focus of the parabola $4y^2+12x-20y+67=0$ is
1 $$\Bigg{(}-\frac{5}{4},\frac{17}{2}\Bigg{)}$$ 2 $$\Bigg{(}-\frac{17}{2},\frac{5}{4}\Bigg{)}$$ 3 $$\Bigg{(}-\frac{17}{4},\frac{5}{2}\Bigg{)}$$ 4 $$\Bigg{(}-\frac{5}{2},\frac{17}{4}\Bigg{)}$$ Go to Discussion
Solution Qus : 5 4 An equilateral triangle is inscribed in the parabola $y^{2} = 4ax$, such that one of the vertices of the triangle
coincides with the vertex of the parabola. The length of the side of the triangle is:
1 $a\sqrt{3}$ 2 $2a\sqrt{3}$ 3 $4a\sqrt{3}$ 4 $8a\sqrt{3}$ Go to Discussion
Solution Qus : 6 2 The locus of the mid points of all chords of the parabola $y^{2}=4x$
which are drawn through its
vertex, is
1 $y^{2}=8x$ 2 $y^{2}=2x$ 3 $$x^{2}+4y^{2}=16$$ 4 $x^{2}=2y$ Go to Discussion
Solution Qus : 7 3 If $x = 1$ is the directrix of the parabola $y^{2} = kx - 8$, then k is:
1 $\frac{1}{8}$ 2 $8$ 3 $4$ 4 $\frac{1}{4}$ Go to Discussion
Solution Qus : 8 4 A normal to the curve $x^{2} = 4y$ passes through the point (1, 2). The distance of the origin from the
normal is
1 $\sqrt{2}$ 2 $2\sqrt{2}$ 3 $\frac{1}{\sqrt{2}}$ 4 $\frac{3}{\sqrt{2}}$ Go to Discussion
Solution Qus : 9 2 The equation of the tangent at any point of curve $x=a cos2t, y=2\sqrt{2} a sint$ with $m$ as its slope is
1 $$y=mx+a(m-\frac{1}{m})$$ 2 $$y=mx-a(m+\frac{1}{m})$$ 3 $$y=mx+a(a+\frac{1}{a})$$ 4 $$y=amx+a(m-\frac{1}{m})$$ Go to Discussion
Solution Qus : 10 2
The locus of the mid-point of all chords of the parabola $y^2 = 4x$ which are drawn through its vertex is
1 $$y^2=8x$$ 2 $$y^2=2x$$ 3 $$x^2+4y^2=16$$ 4 $$x^2=2y$$ Go to Discussion Solution
Locus of Midpoint of Chords
Given Parabola: \( y^2 = 4x \)
Condition: Chords pass through the vertex \( (0, 0) \)
Let the other end of the chord be \( (x_1, y_1) \), so the midpoint is:
\( M = \left( \frac{x_1}{2}, \frac{y_1}{2} \right) = (h, k) \)
Since the point lies on the parabola: \( y_1^2 = 4x_1 \)
⇒ \( (2k)^2 = 4(2h) \)
⇒ \( 4k^2 = 8h \)
⇒ \( \boxed{k^2 = 2h} \)
✅ Locus of midpoints:
\( y^2 = 2x \)
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