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List of top Mathematics Questions on Tangents and Normals asked in JEE Main

Two lines \(L_1 \;\& \;L_2\) passing through origin trisecting the line segment intercepted by the line \(4x + 5y = 20\) between the coordinate axes. Then the tangent of angle between the lines \(L_1\) and \(L_2\) is
If A is the area in the first quadrant enclosed by the curve \(C: 2x^2 – y + 1 = 0\), the tangent to C at the point (1, 3) and the line x + y = 1, then the value of 60A is ________
Let a curve y = f(x), x ∈ (0, ∞) pass through the points  \(p ( 1\frac{3 }{2} )\) and  Q \( ( a,\frac{1 }{2} )\). If the tangent at any point R(b, f(b)) to the given curve cuts the y-axis at the point S (0, c) such that bc = 3, then (PQ)2 is equal to ______.
The tangents at the point $A (1,3)$ and $B (1,-1)$ on the parabola $y^2-2 x-2 y=1$ meet at the point $P$ Then the area (in unit $^2$ ) of $\triangle PAB$ is :-

The number of points on the curve \(y=54 x^5-135 x^4-70 x^3+180 x^2+210 x\) at which the normal lines are parallel \(to x+90 y+2=0\) is 

Let a circle $C_1$ be obtained on rolling the circle $x^2+y^2-4 x-6 y+11=0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$ Let $C_2$ be the image of $C_1$ in $T$ Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:
Let a common tangent to the curves y2 = 4x and (x – 4)2 + y2 = 16 touch the curves at the points P and Q. Then (PQ)2 is equal to ________.
Let m1 and m2 be the slopes of the tangents drawn from the point P(4,1) to the hyperbola H : \(\frac{y^2}{25} − \frac{x^2}{16 }\)= 1. If Q is the point from which the tangents drawn to H have slopes |m1| and |m2| and they make positive intercepts α and β on the x-axis, then  \(\frac{( P Q ) ^2}{ α β}\) α β is equal to _____.
A light ray emits from the origin making an angle \(30\degree\) with the positive x-axis. After getting reflected by the line \(x + y = 1\), if this ray intersects x-axis at Q, then the abscissa of Q is
Let a tangent to the curve $9 x^2+16 y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is
If the tangents at the points P and Q on the circle x2 + y2 – 2x + y = 5 meet at the point R  \(( \frac{9}{4},2 ) \)then the area of the triangle PQR is 
Let the tangent to the curve x2 + 2x – 4y + 9 = 0 at the point P(1, 3) on it meet the y-axis at A. Let the line passing through P and parallel to the line x – 3y = 6 meet the parabola y2 = 4x at B. If B lies on the line 2x – 3y = 8. then (AB)2 is equal to___
Let the tangent at any point P on a curve passing through the points (1, 1) and (\(\frac{1}{10}\),100), intersect positive x-axis and y-axis at the points A and B respectively. If PA : PB = 1 : k and y = y(x) is the solution of the differential equation  \(\frac{dy}{e^{dx}}Kx+\frac{K}{2}\) , y(0) = k, then 4y(1) – 5loge3 is equal to ______ .
The area of the region enclosed by the curve y = x3 and its tangent at the point (-1, -1) is
Let the tangent and normal at the point \((3\sqrt{3},1)\) on the ellipse \(\frac{ x^2}{36}+\frac{y^2}{4} =1\) meet the y-axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line x = \(2\sqrt{5}\) intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point (α,β) , then α2 - β2 is equal to 
Consider the plane 2x + y – 3z = 6. If (𝛼, 𝛽, 𝛾) is the image of point (2, 3, 5) in the given plane, then 𝛼 + 𝛽 + 𝛾 = ______
The number of common tangents, to the circles x² x²+y²-18x-15y+131=0 2 and x2 + y²-6x-6y-7=0, is
The slope of tangent at any point (x, y) on a curve y = y(x) is \(\frac{x^2+y^2}{2xy},x>0\) If y(2) = 0, then a value of y(8) is
 
Let the tangents at the points A(4, –11) and B(8, –5) on the circle \(x^2+y^2-3x+10y-15=0\), intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to

Let S be the set of all the natural numbers, for which the line 
\(\frac{x}{a}+\frac{y}{b}=2 \)
is a tangent to the curve
\((\frac{x}{a})^n+(\frac{y}{b})^n=2 \)
at the point (a, b), ab ≠ 0. Then :

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tanx(cosx – y). If the curve passes through the point (π/4, 0) then the value of 
\(\int_{0}^{\frac{\pi}{2}} y \,dx\)
is equal to :

Let the locus of the centre (α, β), β> 0, of the circle which touches the circle x2 +(y – 1)2 = 1 externally and also touches the x-axis be L. Then the area bounded by L and the line y = 4 is :

Let the function f(x) = 2x2 – logex, x> 0, be decreasing in (0, a) and increasing in (a, 4). A tangent to the parabola y2 = 4ax at a point P on it passes through the point (8a, 8a –1) but does not pass through the point (-1/a, 0). If the equation of the normal at P is
\(\frac{x}{α}+\frac{y}{β}=1\)
 then α + β is equal to _______ .

the angle of intersection of the curves y=x2 and x=y2 at (1,1) is
A function y = f(x) has a second order derivative f''(x) = 6(x - 1).  If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x -5.  then the function is :

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